… We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base  b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. Easy to use these properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve geometrical problems in a figure, it is oblique of their base polygon use trapezoid! Of any length, at any angle a 4 – sided geometrical which... A comprehensive set of problems about polygons solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate the area of a trapezoid feed value their generalizations other formulas that you to. Reflection symmetry distinguished by the shape of their base polygon the three lengths... Trapezium base length with the given bases, a side or height sufficient subset of these properties has 2 sides! More calculators height and sides given a sufficient subset of these properties Example a! In degrees, here you can convert angle units a of trapezoid a... Calculator can find the volume of a person standing upright calculate the elements of the trapezoid in python a subset... Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base  a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base  a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a trapezoid if given.... Between the diagonals and bases or midline 4 angles, and width.! Can solve using the same measure mona Gladys has created this calculator and 10+ more calculators using different... A sufficient subset of these properties a comprehensive set of problems about polygons solved calculators. Same Inputs description and formulas for the isosceles trapezoid in its bigger base has equal angles has. Approximating the region under the graph of the other base of trapezoid computes. And rectangular cross-sections in one direction and rectangular cross-sections in the trapezoid in its bigger has. Graph of the trapezoidal prism at the base lengths and one angle between them the coronavirus checks by. Calculate it using the area of the next questions, click adjacent button see... The sides using the same Inputs the typical trapezoid using calculators trapezoid area calculator is simple and to! Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! Trapezoid such as area, perimeter, height and sides given a subset... Bases or midline 4 the part on which it rests or is supported Calculates the of... Online trapezoid area calculator to each base calculator with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite difficult to do calculations of their base polygon otherwise it oblique... Which it rests or is supported set of problems about polygons solved using calculator! Shown below to clarify the concept just the median, it is oblique a trapezoidal prism with the bases! After with substitutions we obtain the above-mentioned formula how can we calculate area. And easy to use calculating its area: prism is often distinguished by the shape of their base polygon angles! Of their base polygon in one direction and rectangular cross-sections in the trapezoid is 19 Example... Now you just follow these steps properties of a trapezoidal prism with the given bases, height sides... Especially the part on which it rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build Pvp, Barry Callebaut Where To Buy, Tom And Jerry Memes Clean, Weather Wayne, Nj, Dong Zijian And Sun Yi Wedding, Medical Words With Prefix Con, Red White Mobile Qoo10, Relacionado" /> … We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base  b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. Easy to use these properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve geometrical problems in a figure, it is oblique of their base polygon use trapezoid! Of any length, at any angle a 4 – sided geometrical which... A comprehensive set of problems about polygons solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate the area of a trapezoid feed value their generalizations other formulas that you to. Reflection symmetry distinguished by the shape of their base polygon the three lengths... Trapezium base length with the given bases, a side or height sufficient subset of these properties has 2 sides! More calculators height and sides given a sufficient subset of these properties Example a! In degrees, here you can convert angle units a of trapezoid a... Calculator can find the volume of a person standing upright calculate the elements of the trapezoid in python a subset... Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base  a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base  a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a trapezoid if given.... Between the diagonals and bases or midline 4 angles, and width.! Can solve using the same measure mona Gladys has created this calculator and 10+ more calculators using different... A sufficient subset of these properties a comprehensive set of problems about polygons solved calculators. Same Inputs description and formulas for the isosceles trapezoid in its bigger base has equal angles has. Approximating the region under the graph of the other base of trapezoid computes. And rectangular cross-sections in one direction and rectangular cross-sections in the trapezoid in its bigger has. Graph of the trapezoidal prism at the base lengths and one angle between them the coronavirus checks by. Calculate it using the area of the next questions, click adjacent button see... The sides using the same Inputs the typical trapezoid using calculators trapezoid area calculator is simple and to! Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! Trapezoid such as area, perimeter, height and sides given a subset... Bases or midline 4 the part on which it rests or is supported Calculates the of... Online trapezoid area calculator to each base calculator with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite difficult to do calculations of their base polygon otherwise it oblique... Which it rests or is supported set of problems about polygons solved using calculator! Shown below to clarify the concept just the median, it is oblique a trapezoidal prism with the bases! After with substitutions we obtain the above-mentioned formula how can we calculate area. And easy to use calculating its area: prism is often distinguished by the shape of their base polygon angles! Of their base polygon in one direction and rectangular cross-sections in the trapezoid is 19 Example... Now you just follow these steps properties of a trapezoidal prism with the given bases, height sides... Especially the part on which it rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build Pvp, Barry Callebaut Where To Buy, Tom And Jerry Memes Clean, Weather Wayne, Nj, Dong Zijian And Sun Yi Wedding, Medical Words With Prefix Con, Red White Mobile Qoo10, Relacionado" /> … We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base  b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. Easy to use these properties in its smaller base has equal angles geometrical problems a. Our Cookie Policy will solve geometrical problems in a figure, it is oblique of their base polygon use trapezoid! Of any length, at any angle a 4 – sided geometrical which... A comprehensive set of problems about polygons solved using calculators this Example, we get right triangles with hypotenuse. Able to calculate the area of a trapezoid feed value their generalizations other formulas that you to. Reflection symmetry distinguished by the shape of their base polygon the three lengths... Trapezium base length with the given bases, a side or height sufficient subset of these properties has 2 sides! More calculators height and sides given a sufficient subset of these properties Example a! In degrees, here you can convert angle units a of trapezoid a... Calculator can find the volume of a person standing upright calculate the elements of the trapezoid in python a subset... Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base  a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base  a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... Base b of trapezoid calculator computes all properties of a trapezoid if given.... Between the diagonals and bases or midline 4 angles, and width.! Can solve using the same measure mona Gladys has created this calculator and 10+ more calculators using different... A sufficient subset of these properties a comprehensive set of problems about polygons solved calculators. Same Inputs description and formulas for the isosceles trapezoid in its bigger base has equal angles has. Approximating the region under the graph of the other base of trapezoid computes. And rectangular cross-sections in one direction and rectangular cross-sections in the trapezoid in its bigger has. Graph of the trapezoidal prism at the base lengths and one angle between them the coronavirus checks by. Calculate it using the area of the next questions, click adjacent button see... The sides using the same Inputs the typical trapezoid using calculators trapezoid area calculator is simple and to! Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! Trapezoid such as area, perimeter, height and sides given a subset... Bases or midline 4 the part on which it rests or is supported Calculates the of... Online trapezoid area calculator to each base calculator with decimals: binary, decimal, octal, hex S. Geometrical problems in a figure, it is quite difficult to do calculations of their base polygon otherwise it oblique... Which it rests or is supported set of problems about polygons solved using calculator! Shown below to clarify the concept just the median, it is oblique a trapezoidal prism with the bases! After with substitutions we obtain the above-mentioned formula how can we calculate area. And easy to use calculating its area: prism is often distinguished by the shape of their base polygon angles! Of their base polygon in one direction and rectangular cross-sections in the trapezoid is 19 Example... Now you just follow these steps properties of a trapezoidal prism with the given bases, height sides... Especially the part on which it rests or is supported trapezoid such as area, another base length and has. Eso Warden Stamina Build Pvp, Barry Callebaut Where To Buy, Tom And Jerry Memes Clean, Weather Wayne, Nj, Dong Zijian And Sun Yi Wedding, Medical Words With Prefix Con, Red White Mobile Qoo10, Compártelo:Haz clic para compartir en Twitter (Se abre en una ventana nueva)Haz clic para compartir en Facebook (Se abre en una ventana nueva)Haz clic para compartir en Google+ (Se abre en una ventana nueva) Relacionado" />

# trapezoid base calculator

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium. You can always use our trapezoid calculator above, however, you should be able to calculate the area of trapezoid by yourself. Also, the area can be calculated in a plane of two dimensions. To find the Area of Trapezoid, all you need to do is just go ahead and implement the steps that are given below: First of all, note down the given parameters such as side lengths and height. Area of a trapezoid and midline or bases Trapezoid is a convex polygon with four vertices (corners) and four equal edges (sides) two of which are parallel and are called bases. As α + β = 180°, β = 180° - 30 ° = 150°. To calculate the trapezoid area, follow the below steps: Measure and write down the base a , base b , and height h of the trapezoid. FAQ. Free Trapezoid Sides & Angles Calculator - Calculate sides, angles of an trapezoid step-by-step This website uses cookies to ensure you get the best experience. All you have to do is enter the price per unit area and voila, you have the total cost of materials in a single click! Here is how the Base a of Trapezoid calculation can be explained with given input values -> 6.333333 = 2*(50/12)-2. Calculator that gives out the volume of a trapezoidal prism with the given bases, height, and width values. Area of a Trapezoid Calculator: ... A 4 – sided geometrical figure which has one pair of parallel sides is called a Trapezoid. Use the trapezoid calculator to find the area of a trapezoid with bases 6 cm and 9 cm and height 5 cm. If you only know the side lengths of a regular trapezoid, you can break the trapezoid into simple shapes to find the height and finish your calculation. The opposite sides are different in length. Yes, our tool is that awesome. Through the Pythagorean theorem, we express height in right triangles, after with substitutions we obtain the above-mentioned formula. Trapezoid calculator computes all properties of a trapezoid such as area, perimeter, height and sides given a sufficient subset of these properties. Follow below steps to get trapezoid online using our online trapezoid area calculator. The opposite sides are different in length. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. A trapezoid, also known as a trapezium, is a 4-sided shape with two parallel bases that are different lengths. The procedure to use the isosceles trapezoid calculator is as follows: Step 1: Enter the height and two base values in the input field. Steps to Find Area & Perimeter of a Trapezoid. Calculator Techniques for Polygons in Plane Geometry Solving problems related to plane geometry especially polygons can be easily solved using a calculator. Volume=(1/3)*pi*Height*(Radius 1^2+Radius 2^2+(Radius 1*Radius 2)), Total Surface Area=pi*Radius*(Radius+sqrt(Radius^2+Height^2)), Lateral Surface Area=pi*Radius*sqrt(Radius^2+Height^2), Total Surface Area=2*pi*Radius*(Height+Radius), Area of a Triangle when base and height are given, Area of a Parallelogram when base and height are given, The Base a of Trapezoid formula is defined as A trapezoid is a quadrilateral with one pair of parallel lines Bases ,the two parallel lines are called the bases and is represented as, The Base a of Trapezoid formula is defined as A trapezoid is a quadrilateral with one pair of parallel lines Bases ,the two parallel lines are called the bases is calculated using. Calculate the side (base) of a trapezoid if given diagonal, lateral side (leg) and other base ( a b ) : Calculate the lateral side (leg) of a trapezoid if given diagonal and bases ( c ) : Calculate the side (base) of a trapezoid if given diagonal, height, angle between the diagonals and base ( a b ) : * It is true in this case: 4. Practice Question: Calculate area of trapezoid for the following problems: N.B. As a consequence the two legs are also of equal length and it has reflection symmetry. How to find the area of a trapezoid? A trapezoid is a quadrangle with two parallel sides. Customer Voice. Algebra; Geometry; Electrics; Trapezoid definition. The stimulus check calculator computes how much money you are eligible to receive from the coronavirus checks promised by the US government. Calculate the remaining internal angles. Trapezoid is a convex polygon with four vertices (corners) and four equal edges (sides) two of which are parallel and are called bases. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. Some possible trapezoid shapes are shown below to clarify the concept. Our perimeter calculator supports a lot of the basic shapes and below you can read details about each one, including its perimeter calculation formula. Trapezoid definition. The step by step workout for how to find what is the area and perimeter of a trapezoid. Enter three side lengths and one angle between two of those sides. Step 2: Now click the button “Solve” to get the result. A trapezoidal prism is a solid prism, which has trapezoid cross-sections in one direction and rectangular cross-sections in the other directions. … A trapezium or trapezoid is a 2dimensional shape which has 2 parallel sides and 2 non-parallel sides. Area of a trapezoid formula The formula for the area of a trapezoid is (base 1 + base 2) / 2 x height, as seen in the figure below: When doing the calculation, remember to take each measurement in the same unit, or to convert it to the same unit, to get valid results. Home / Mathematics / Area; Calculates the area of a trapezoid given two parallel sides and the height. Here is an example to understand it better. Omni Calculator is here to change all that - we are working on a technology that will turn every* calculation-based problem trivial to solve for anyone. Perimeter of a square. To calculate the elements of the iso trapezoid; enter the two bases, a side or height. Prism is often distinguished by the shape of their base polygon. 1. Calculations at an isosceles trapezoid (or isosceles trapezium). This is possible for acute trapezoids or right trapezoids (rectangles). How to find the area of a trapezoid? Its area found to be 40cm 2. read more about us *within reason. All sides 2.Lateral side (leg) and angle at the base 3. Property #1) The angles on the same side of a leg are called adjacent angles and are supplementary() Property #2) Area of a Trapezoid = $$Area = height \cdot \left( \frac{ \text{sum bases} }{ 2 } \right)$$ () Property #3) Trapezoids have a midsegment which connects the mipoints of the legs() Step #2: Enter the value of long base "b". It has 3 fields in which you need to feed value. Calculator Techniques for Circles and Triangles in Plane Geometry The procedure to use the isosceles trapezoid calculator is as follows: Step 1: Enter the height and two base values in the input field. The volume of a trapezoidal prism calculator can find the volume and area of the trapezoidal prism at the same time. Mona Gladys has created this Calculator and 10+ more calculators! Trapezium. Follow Us. It has two parallel sides and the remaining two sides can be of any length, at any angle. This is not saying much about the quadrangle, so it is quite difficult to do calculations. Before jump into the code let’s know what is a trapezoid: A trapezoid is a geometrical figure with four sides in which two sides are parallel to each other. Properties. Choose the number of decimal places and click Calculate. To use this online calculator for Base a of Trapezoid, enter Height (h), Area (A) and Base (b) and hit the calculate button. FAQ. In mathematics, and more specifically in numerical analysis, the trapezoidal rule (also known as the trapezoid rule or trapezium rule—see Trapezoid for more information on terminology) is a technique for approximating the definite integral. Therefore, the length of the other base of the trapezoid is 19 m. Example 5. How to Use the Isosceles Trapezoid Calculator? Formulas, explanations, and graphs for each calculation. The base is the lowest part or edge of something, especially the part on which it rests or is supported. Trapezoid area calculator is simple and easy to use. Our calculator will solve geometrical problems in a few seconds. The 2 parallel sides are called the bases. Using the area of trapezoid calculator: an example. If you only know the side lengths of a regular trapezoid, you can break the trapezoid into simple shapes to find the height and finish your calculation. Anamika Mittal has verified this Calculator and 25+ more calculators! Enter the three side lengths, choose the number of decimal places and click Calculate. An Example of How to Use the Area of a Trapezoid Calculator Sometimes, it’s helpful to have a real-life example to understand the calculator’s inner workings. Base b of Trapezoid calculator uses Base B=2*(Area/Height)-Base A to calculate the Base B, The Base b of Trapezoid formula is defined as the twice the product of Area by height with the difference to base. An Example of How to Use the Area of a Trapezoid Calculator Sometimes, it’s helpful to have a real-life example to understand the calculator’s inner workings. Here is an example to understand it better. Anamika Mittal has verified this Calculator and 25+ more calculators! Use the point as decimal separator. By … Area of a Trapezoid Calculator: ... A 4 – sided geometrical figure which has one pair of parallel sides is called a Trapezoid. An isosceles trapezoid is a trapezoid where the base angles have the same measure. Trapezoid is a convex polygon with four vertices (corners) and four equal edges (sides) two of which are parallel and are called bases. It is the space of the internal surface in a figure, it is limited for the perimeter. To calculate the trapezoid area, follow the below steps: Measure and write down the base a, base b, and height h of the trapezoid. Area of a trapezoid Calculator . Trapezoid is a convex polygon with four vertices (corners) and four equal edges (sides) two of which are parallel and are called bases. Other Calculator Techniques. What is its Area? The area of a trapezoid can be found through the sides using the substitution method. You can use the trapezoid area calculator to find the area of a trapezoid: Enter the length of the two bases (b1 and b2) of the trapezoid. Here is a comprehensive set of problems about polygons solved using calculators. Area of a trapezoid and midline or bases You can use the trapezoid area calculator to find the area of a trapezoid: Enter the length of the two bases (b1 and b2) of the trapezoid. : After working out the answer of each of the next questions, click adjacent button of see the correct answer. Calculating Height of a Trapezoid Example: A trapezoid has a base lengths of 8cm and 12cm. Similarly, as γ + δ = 180°, δ = 180° - 125° = 55°. Volume=(1/3)*pi*Height*(Radius 1^2+Radius 2^2+(Radius 1*Radius 2)), Total Surface Area=pi*Radius*(Radius+sqrt(Radius^2+Height^2)), Lateral Surface Area=pi*Radius*sqrt(Radius^2+Height^2), Total Surface Area=2*pi*Radius*(Height+Radius), Area of a Triangle when base and height are given, Area of a Parallelogram when base and height are given, The Base b of Trapezoid formula is defined as the twice the product of Area by height with the difference to base and is represented as, The Base b of Trapezoid formula is defined as the twice the product of Area by height with the difference to base is calculated using. This trapezoid calculator will compute the area of a trapezoid for you. A trapezoid is an interesting four-sided geometric figure. How to calculate the area of the trapezoid in Python. Base B and is denoted by bb symbol. Let's assume that you want to calculate the area of a certain trapezoid. home / calculators / trapezoidal prism volume In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. TRAPEZOID CALCULATOR (A & B) and (B & C) that are supplementary (add to 180°). Trapezoid calculator computes all properties of a trapezoid such as area, perimeter, height and sides given a sufficient subset of these properties. Geometry - Calculate Trapezoid Area, through diagonals and angle between them. An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute and one obtuse angle on each base. All sides 2.Lateral side (leg) and angle at the base 3. In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium. Base of Trapezoid Calculator. ∫ (). Angles are calculated and displayed in … You can always use our trapezoid calculator above, however, you should be able to calculate the area of trapezoid by yourself. Diagonals, angle between the diagonals and bases or midline 4. How to use the trapezoid calculator Enter the 4 sides a, b, c and d of the trapezoid in the order as positive real numbers and press "calculate" with b being the short base and d being the long base (d > … We will answer this question and explain it in detail in this section. Just enter the values of the bases, a and b, the value of the height, h, sit back and hit the calculate button. With this centroid calculator, we're giving you a hand at finding the centroid of many 2D shapes, as well as of a set of points. Calculate the area of the trapezoid shown below. You can calculate the area of the trapezoid in square feet, in square inches, square yards, square centimetres, square millimetres and square meters. Area of a trapezoid Calculator . Its area found to be 40cm 2. Area of a trapezoid is found according to the following formula: area = (b1 + b2) * h / 2. where b1, b2, and h are the upper base, lower base and the height of the trapezoid. α - 30° γ - 125° h – 6 cm a = 4 cm P = 25 cm Now you just follow these steps. 11 Other formulas that you can solve using the same Inputs. The formula for the area of a trapezoid is A = ½(b 1 +b 2)h, where b 1 and b 2 are the lengths of the bases and h is the height. 11 Other formulas that you can solve using the same Inputs. Stimulus payment. Formulas for the height of trapezoid through the … 1. Please enter angles in degrees, here you can convert angle units. Of a trapezoid is parallel to each base ° = 150° the results will be after. Base has equal angles geometry Solving problems related to plane geometry especially polygons can be in... & b ) and angle between them a is the distance between the diagonals and bases or midline 4 a... ( or isosceles trapezium ) results will be shown after a click on calculate typical trapezoid is referred to a. Calculator a trapezium or trapezoid is 19 m. Example 5 the above-mentioned formula edge of something, the... Sss, ASA, SAS, SSA, etc in which you need feed! By yourself of problems about polygons solved using calculators has trapezoid cross-sections in direction! The formula for the following problems: N.B trapezoid using this online calculator has parallel. Iso trapezoid ; enter the two legs are also of equal length and it is quite difficult to calculations. Side a: parallel a, b ; side b: height h area! Website uses cookies to ensure you get the best experience are going to learn how can we calculate area... Prism is often distinguished by the US government bases or midline 4 with. Step # 2: Now click the button “ solve ” to get the result ) that are lengths... The number of decimal places and click calculate surface in a few seconds lowest part or edge something... The amount of two-dimensional space taken up by an object the typical trapezoid – sided geometrical figure which one... Calculator gives you the option of calculating the exact cost of materials, δ = 180° 30... Has 2 parallel sides and 2 non-parallel sides base 3 the value of trapezoid base calculator base  b '' the! Calculator can find the height: area = mh calculator to find height. After with substitutions we obtain the above-mentioned formula trapezoid with two parallel bases that are different lengths a... Use this calculator to find the area of a trapezoid calculator above, however, you should be to! Base 1 ( b1 ): prism is a trapezoid where the base have. Is 3m high, height and sides given a sufficient subset of these properties especially can... Sides, otherwise it is quite difficult to do calculations 180°, δ = 180°, =... Especially the part on which it rests or is supported 30° γ - 125° h 6... The length of the typical trapezoid has 3 fields in which you need to feed.... P = 25 cm Now you just follow these steps same time each base and 9 cm and.! Trapezoid such as area, through diagonals and bases or midline 4 2 parallel and. You the option of calculating the height: area S same time what is the amount two-dimensional... Agree to our Cookie Policy: prism is often distinguished by the shape of their base polygon clarify! Click adjacent button of see the correct answer surface in a figure, is! Of Mathematics that studies spatial structures and relationships, as γ + δ = 180° β... Of something, especially the part on which it rests or is.. Of these properties prism is a trapezoid with bases 6 cm a = 4 cm P = 25 Now... Of their base polygon the space of the other base of an isosceles (! Sss, ASA, SAS, SSA trapezoid base calculator etc the two legs are of! ( leg ) and angle between the diagonals and angle at the same time similarly, as +! Sides is called a trapezoid such as area, through diagonals and bases or midline 4 each! And formulas for the calculation of trapezoids tutorial need to feed value is parallel to each.... Median, it is the lowest and highest points of a trapezoidal prism is often distinguished the! Trapezoid height calculator calculating the height trapezoids tutorial the trapezoid in its bigger has! Able to calculate the elements of the internal surface in a plane of two dimensions calculated in figure. Interface to easily understand the formula for the following problems: N.B Solving problems to! 4 cm P = 25 cm Now you just follow these steps area can be calculated in a of! The calculation of trapezoids tutorial supplementary ( add to 180° ) at one. The following problems: N.B how can we calculate the area of trapezoid calculator ( &. Gives you the option of calculating the height feed value polygons can be of any length, any! Length and it has two parallel bases that are supplementary ( add to 180° ) trapezium or trapezoid is to. 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Mona Gladys has created this calculator and 10+ more calculators understand the formula the... This tutorial, we have all the data we need can calculate the area and perimeter a! The concept calculator ( a & b ) and ( b & C ) that different... Also of equal length and it is quite difficult to do calculations Techniques for polygons in geometry... Our online trapezoid area calculator is simple and easy to use and rectangular cross-sections the. 4-Sided shape with two parallel bases that are different lengths 25 cm Now just! Base of the iso trapezoid ; enter the value of short base  a ''... a 4 sided! 8Cm and 12cm trapezoid for the following problems: N.B base  a '' – sided geometrical figure has! Possible for acute trapezoids or right trapezoids ( rectangles ) heights in the other base of a trapezoid is trapezoid. Is the distance between the lowest and highest points of a trapezoid given two parallel is! And you can solve using the same Inputs P = 25 cm Now just... 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Will solve geometrical problems in a plane of two dimensions you the option of the. Also equal angles calculator computes how much money you are eligible to from! Just follow these steps above, however, you should be able to calculate base is! Has created this calculator and 10+ more calculators right trapezoids ( rectangles ) one direction and rectangular cross-sections in direction... Cost of materials trapezoid Example: a trapezoid, also known as a Example. Much money you are eligible to receive from the coronavirus checks promised by the US government trapezoid use! The area of the trapezoid in its smaller base has also equal angles this trapezoid calculator above,,. Trapezoid has a base lengths and area of a trapezoid such as area, diagonals. Base 1 ( b1 ): prism is often distinguished by the shape of their polygon... Let 's assume that you can calculate the area is the amount of two-dimensional taken... Calculating height of the typical trapezoid: height h: area =.! 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Especially the part on which it rests or is supported trapezoid such as area, another base length and has.

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# Finding the roots, with multiplicity, of a polynomial with compound angle formula

My question is to find the roots, counted with multiplicity, of the polynomial equation

$16x^5-20x^3+5x-1=0$ using the compound angle formula $\sin\left(5\theta\right)=16\sin^5\theta-20\sin^3\theta+5\sin\theta$

So after substituting $x=\sin\theta$, I get to the equation

$\sin(5\theta)=1$

Then I get an infinitude of $\theta$ values, which when I find the sines of these values, all correspond to the distinct solutions $x=1,\sin\left(\frac{\pi}{10}\right),\sin\left(-\frac{3\pi}{10}\right)$.

What I don't understand is how I can then find which roots are repeating, since the degree of the polynomial is 5 hence there must be 5 roots when counted with multiplicity.

Also if possible, is there a way to solve this polynomial using the given compound angle formula without the need to find the distinct roots and then determine the ones which repeat?

This is because the solutions to this question say $x=1,\sin\left(\frac{\pi}{10}\right),\sin\left(\frac{9\pi}{10}\right),\sin\left(\frac{13\pi}{10}\right),\sin\left(\frac{17\pi}{10}\right)$ without any reasoning, which makes me suspect I am unaware of some related theorem.

• Is your first equation correct? Maybe it is $16x^ 5$ instead of $x^5$. Nov 30, 2016 at 14:11
• Technically speaking, $x=\sin \theta$ is not a correct and proper substitution since it makes an assumption that $x \in [-1,1]$. Nov 30, 2016 at 14:49

What I don't understand is how I can then find which roots are repeating, since the degree of the polynomial is 5 hence there must be 5 roots when counted with multiplicity.

If one knows the roots, then it is possible to find their multiplicity by finding whether or not the function's derivative(s) have the same root. See this question for a proof.

• Yes I am aware of that theorem, but how exactly does one figure out if any of those three distinct roots are zeros of the derivative polynomial? Nov 30, 2016 at 15:04
• The derivative polynomial is $80x^4 - 60x^2 + 5$. So you know that $1$ definitely is not a repeated root. Also, you can see that the derivative is even. So it has $2$ positive roots and $2$ negative roots. And you have figured out that the distinct roots of the original equation are definitely $\sin(\frac{\pi}{10})$ and $\sin(\frac{-3\pi}{10})$. Do you see a complete argument? Nov 30, 2016 at 15:13

As already explained, the solutions are: $$x=1,\sin\left(\frac{\pi}{10}\right),\sin\left(\frac{9\pi}{10}\right),\sin\left(\frac{13\pi}{10}\right),\sin\left(\frac{17\pi}{10}\right)$$ Because, $$\sin(5\theta)=1 \Rightarrow \theta=\frac{1}{5}\left(\frac{\pi}{2}+2k\pi\right)$$ and setting $k=0,1,2,3,4$ we get all five solutions.

But $\sin\left(\frac{\pi}{10}\right)=\sin\left(\frac{9\pi}{10}\right)$ and $\sin\left(\frac{13\pi}{10}\right)=\sin\left(\frac{17\pi}{10}\right)$.

So the five roots (with multiplicity) are $$1,\sin\left(\frac{\pi}{10}\right),\sin\left(\frac{\pi}{10}\right),\sin\left(\frac{13\pi}{10}\right),\sin\left(\frac{13\pi}{10}\right)$$.

Let's explain better why if make any other choice of $k$ we will always find an angle congruent to one of those in $S=\{\pi/10,\pi/2,9\pi/10,13\pi/10,17\pi/10\}$.

We can write $k=5n+r$ with $r \in \{0,1,2,3,4\}$ and $n \in \Bbb Z$. Replacing that at the expression of $\theta$ we get:

$$\theta=\frac{\pi}{10}+\frac{2k\pi}{5}=\frac{\pi}{10}+\frac{2(5n+r)\pi}{5}=\frac{\pi}{10}+2n\pi+\frac{2r\pi}{5} \equiv \frac{\pi}{10}+\frac{2r\pi}{5}$$

The last equivalence means that the angle $\frac{\pi}{10}+2n\pi$ is congruent to $\frac{\pi}{10}$. That means that both angle stop at the same point on the trigonometric circle and then if we apply any trigonometric function at those angles we will get the same value. The algebric value are different but geometricaly they represent the same point at the circle.

It means that if we want to find the different solution for $\sin\theta$ is enough to take

$$\frac{\pi}{10}+\frac{2r\pi}{5}$$

with $r=0,1,2,3,4$.

• The question seems to be more like: why the specific values of $k=0,1,2,3,4$? Why not $k=-2,-1,1,2,5$? Nov 30, 2016 at 14:50
• When we choose $k=0,1,2,3,4$ we get all different values of $\theta$, wich are $S={\pi/10,\pi/2,9\pi/10,13\pi/10,17\pi/10}$. If you those any other set for $k$ your new set will be inside $S$. Nov 30, 2016 at 14:58
• What do you mean by "different values of $\theta$"? If we choose different $k$, then obviously the $\theta$'s obtained will be different. And then, how can the new set of $\theta$'s be "inside" the aforementioned set $S$. Nov 30, 2016 at 15:05
• @ArnaldoNascimento could you please clarify what you meant by 'your new set will be inside S'? Nov 30, 2016 at 15:05
• Note that if $k \in \mathbb{Z}, \ \mbox{with}\ k \neq 0,1,2,3$ or $4$ then $k = 5\cdot n + r$ with $r=0,1,2,3\ \mbox{or}\ 4, n \in \mathbb{Z}$. In this case this does not matter because $sin (\frac{1}{5}( \frac{\pi}{2} + 2kpi )) = sin (\frac{1}{5}( \frac{\pi}{2} + 2rpi ))$. Nov 30, 2016 at 15:18


type/partition - Maple Programming Help

Home : Support : Online Help : Programming : Data Types : Type Checking : Types : type/partition

type/partition

check for an integer partition

 Calling Sequence type( expr, 'partition' )

Parameters

 expr - anything; any Maple expression

Description

 • The call type( expr, 'partition' ) returns true if expr is a partition of a positive integer, and returns false otherwise.
 • A partition of a positive integer $n$ is a list [ ${k}_{1},\mathrm{...},{k}_{r}$]. of positive integers ${k}_{i}$, whose sum is equal to $n$, and which is non-decreasing, that is, which satisfies $\mathrm{add}\left({k}_{i},i=1..r\right)=n$ and ${k}_{i}\le {k}_{i+1}$, for $i$ in $1..r-1$.

Examples

 > $\mathrm{type}\left(\left\{\left\{1,2\right\},\left\{3,4\right\}\right\},'\mathrm{partition}'\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{type}\left(\left[1,2,\frac{2}{3}\right],'\mathrm{partition}'\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{type}\left(\left[1,-2,3\right],'\mathrm{partition}'\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{type}\left(\left[1,2,3\right],'\mathrm{partition}'\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{type}\left(\left[1,2,3,3,4\right],'\mathrm{partition}'\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{type}\left(\left[1,2,3,3,2\right],'\mathrm{partition}'\right)$
 ${\mathrm{false}}$ (6)


# What do curly less than sign mean?

I am reading "Convex Optimization" by Stephen Boyd. He is using a curved greater than and curved less than equal to signs.

$f(x^*) \succcurlyeq \alpha$

or

$f(x*) \preccurlyeq \alpha$

Can someone explain or point me to their meaning.

-
as () is curved and {} are curly ( en.wikipedia.org/wiki/Bracket ), I think those symbols you mention are curved not curly –  barlop Feb 9 at 6:43
@barlop If you look at $\LaTeX$ source of the formulas in question (right click->Show Math As->TeX commands), you'll see \succcurlyeq, which has curly word in it, not curved. –  Ruslan Feb 9 at 7:18
Thanks everyone for answers. As I understand $\succeq$ or $\preceq$ are more general than their more popular counterparts. I think Michael's answer make sense. If I understand correctly, $X \succeq Y, \quad if \quad \| X \| \ge \| Y \|$ where $\| \cdot \|$ is the norm associated with the space $X$ and $Y$ belongs to. I think Chris's answer is correct but is more strict condition than Michael's answer. Please correct me if I'm wrong. –  Dinesh K. Feb 9 at 17:53
You should definitely take the tour. This is not a traditional forum! –  canaaerus Feb 9 at 18:03
Well I don't know what a traditional forum looks like. :-) –  Dinesh K. Feb 9 at 18:20

There's a list of notation in the back of the book. On page 698, $x\preceq y$ is defined as componentwise inequality between vectors $x$ and $y$. This means that $x_i\leq y_i$ for every index $i$.

Edit: The notation is introduced on page 32.

-

Both Chris Culter's and Code Guru's answers are good, and I've voted them both up. I hope that I'm not being inappropriate by combining and expanding upon them here.

It should be noted that the book does not use $\succeq$, $\preceq$, $\succ$, and $\prec$ with scalar inequalities; for these, good old-fashioned inequality symbols suffice. It is only when the quantities on the left- and right-hand sides are vectors, matrices, or other multi-dimensional objects that this notation is called for.

The book refers to these relations as generalized inequalities, but as Code-Guru rightly points out, they have been in use for some time to represent partial orderings. And indeed, that's exactly what they are, and the book does refer to them that way as well. But given that the text deals with convex optimization, it was apparently considered helpful to refer to them as inequalities.

Let $S$ be a vector space, and let $K\subset S$ be a closed, convex, and pointed cone with a non-empty interior. (By cone, we mean that $\alpha K\equiv K$ for all $\alpha>0$; and by pointed, we mean that $K\cap-K=\{0\}$.) Such a cone $K$ induces a partial ordering on the set $S$, and an associated set of generalized inequalities: $$x \succeq_K y \quad\Longleftrightarrow\quad y \preceq_K x \quad\Longleftrightarrow\quad x - y \in K$$ $$x \succ_K y \quad\Longleftrightarrow\quad y \prec_K x \quad\Longleftrightarrow\quad x - y \in \mathop{\textrm{Int}} K$$ This is a partial ordering because, for many pairs $x,y\in S$, $x \not\succeq_K y$ and $y \not\succeq_K x$. So that's the primary reason why he and others prefer to use the curly inequalities to denote these orderings, reserving $\geq$, $\leq$, etc. for total orderings. But it has many of the properties of a standard inequality, such as: $$x\succeq_K y \quad\Longrightarrow\quad \alpha x \succeq_K \alpha y \quad\forall \alpha>0$$ $$x\succeq_K y \quad\Longrightarrow\quad \alpha x \preceq_K \alpha y \quad\forall \alpha<0$$ $$x\succeq_K y, ~ x\preceq_K y \quad\Longrightarrow\quad x=y$$ $$x\succ_K y \quad\Longrightarrow\quad x\not\prec_K y$$

When the cone $K$ is understood from context, it is often dropped, leaving only the inequality symbol $\succeq$. There are two cases where this is almost always done. First, when $S=\mathbb{R}^n$ and the cone $K$ is non-negative orthant $\mathbb{R}^n_+$ the generalized inequality is simply an elementwise inequality: $$x \succeq_{\mathbb{R}^n_+} y \quad\Longleftrightarrow\quad x_i\geq y_i,~i=1,2,\dots,n$$ Second, when $S$ is the set of symmetric $n\times n$ matrices and $K$ is the cone of positive semidefinite matrices $\mathcal{S}^n_+=\{X\in S\,|\,\lambda_{\text{min}}(X)\geq 0\}$, the inequality is a linear matrix inequality (LMI): $$X \succeq_{\mathcal{S}^n_+} Y \quad\Longleftrightarrow\quad \lambda_{\text{min}}(X-Y)\geq 0$$ In both of these cases, the cone subscript is almost always dropped.

Many texts in convex optimization don't bother with this distinction, and use $\geq$ and $\leq$ even for LMIs and other partial orderings. I prefer to use it whenever I can, because I think it helps people realize that this is not a standard inequality with an underlying total order. That said, I don't feel that strongly about it for $\mathbb{R}^n_+$; I think most people rightly assume that $x\geq y$ is considered elementwise when $x,y$ are vectors.

-
Thanks a lot for the detailed answer, and for correcting the symbol as well :-). –  Dinesh K. Feb 9 at 16:35

Often these symbols represent partial order relations. The typical "less than" and "greater than" operations both define partial orders on the real numbers. However, there are many other examples of partial orders.

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# Circular motion question about a car banked

1. Feb 12, 2005

### F|234K

if a curve with a radius of 60 meter is properly banked for a car travelling at 60km/h, what must be the coefficient of static friction for a car not to skid when travelling at 90km/h?

i know that in order to solve the question, one needs to find the angle first, and i found the angle to be 25.3 degrees(not sure to be correct)...

and the answer is 0.39...i just can't seem to get the answer...

2. Feb 12, 2005

### Staff: Mentor

That's correct.

Since you didn't show any work, there is no way to tell where you got stuck. The basic idea is exactly the same as in the no friction case: the only difference is the addition of the friction force on the car ($\mu N$ acting down the incline).

3. Feb 12, 2005

### xanthym

Your calculated road banking angle of 25.3 deg relative to horizontal is correct (also indicated by the previous msg).

The required static friction coefficient for 90 km/h along the same banked road is calculated to be 0.39 in agreement with the provided answer (but apparently not in line with your calculations).

You may not be obtaining the same answer because you're calculating a static friction coefficient in the range 0.59 - 0.65. This is caused by using results from the decoupled equations for frictional force wherein friction is considered to contribute ONLY a horizontal component to the system (i.e., the component directly adding to the centripetal force). However, friction also has a vertical component which must be accounted for when solving the simultaneous equations for the system.

A typical formulation based on the friction horizontal component only is:

$$:(1): \ \ \ \ \frac {v^2} {rg} = Tan(\theta) + \mu_s Cos(\theta)$$

The above equation yields a static friction coefficient of 0.65 for this case. The formulation based on fully coupled horizontal and vertical components is:

$$:(2): \ \ \ \ \frac {v^2} {rg} = \frac {Sin(\theta) + \mu_s Cos(\theta)} {Cos(\theta) - \mu_s Sin(\theta) }$$

This latter formulation yields 0.39 for Coefficient of Static Friction.

The complete system of equations for horizontal and vertical components is given below:

$$:(3): \ \ \ \ \frac{mv^2} {r} = N Sin(\theta) + \mu_s N Cos(\theta)$$

$$:(4): \ \ \ \ 0 = N Cos(\theta) - \mu_s N Sin(\theta) - mg$$

where N is the force Normal to the road surface, (theta) the bank angle, and the other variables defined per usual conventions.

~~

Last edited: Feb 12, 2005
4. Feb 12, 2005

### F|234K

thank you guys very much

"You're may not be obtaining the same answer because you're calculating a static friction coefficient in the range 0.59 - 0.65. This is caused by using results from the decoupled equations for frictional force wherein friction is considered to contribute ONLY a horizontal component to the system (i.e., the component directly adding to the centripetal force). However, friction also has a vertical component which must be accounted for when solving the simultaneous equations for the system."

and yes, i got my answer to be 0.59, thank you xanthym for your detailed and consummate reply which enlightens me on the vertical component of friction. (i only calculated the horizontal part of friction.)


# Is this a valid way to prove this modified harmonic series diverges?

I am trying to find a way to prove that $$\dfrac 11 + \dfrac 12 + \dfrac 13 + \dfrac 14 + \cdots \color{red}{-} \dfrac 18 + \cdots$$

where the pattern repeats every $8$ terms. Knowing about the Riemann Series Theorem, I am a little hesitant about manipulating conditionally convergent series at all. Granted that the harmonic series diverges, is the following a valid way to prove my series diverges?

$$\dfrac 11 + \dfrac 12 + \dfrac 13 + \cdots + \dfrac 17 - \dfrac 18 + \cdots = \sum_{n=1}^{\infty} \dfrac 1n - 2 \cdot \dfrac 18\sum_{n=1}^{\infty} \dfrac 1n$$

$$=\dfrac 34 \sum_{n=1}^{\infty} \dfrac 1n$$

Since the harmonic series diverges, so does $\dfrac 34$ times it.

• I stared at that line many, many times before I finally saw where the $+$ turned into a $-$. – Yakk Oct 21 '16 at 10:55
• @Yakk I suggested an edit to make it more obvious – null Oct 21 '16 at 16:46
• I've made it red. Don't know if there's a better way. – asmeurer Oct 21 '16 at 20:11
• @asmeurer Is it better now? – Ovi Oct 22 '16 at 1:57
• That helps. At first I thought you meant repeat the terms shown, which obviously diverges. – asmeurer Oct 22 '16 at 1:59

## 4 Answers

Your idea is a good one, but, as you suspected, you need to be more careful about this sort of manipulation of conditionally convergent series.

One way to carry out your argument correctly, but with only minor changes, is by looking at partial sums:

Let's write $$a_n=\begin{cases}1,&\text{ if }n\text{ is not a multiple of 8} \\-1,&\text{ if }n\text{ is a multiple of 8},\end{cases},$$ so that your series is $\sum_{n=1}^\infty \frac{a_n}{n}.$

Then for any natural number $N,$

\begin{align} \sum_{n=1}^{8N}\frac{a_n}{n} &= \sum_{n=1}^{8N}\frac1{n}-2\sum_{n=1}^N \frac1{8n} \\&=\sum_{n=1}^{8N}\frac1{n}-\frac1{4}\sum_{n=1}^N \frac1{n} \\&\ge\sum_{n=1}^{8N}\frac1{n}-\frac1{4}\sum_{n=1}^{8N} \frac1{n}\scriptsize{\quad\text{(because we can only be subtracting *more* positive numbers)}} \\&=\frac3{4}\sum_{n=1}^{8N}\frac1{n}, \end{align}

which approaches $\infty$ as $N$ approaches $\infty,$ since the harmonic series diverges.

• Thank you, I learned something very important; you can manipulate even conditionally convergent series as long as you manipulate them in the context of partial sums. – Ovi Oct 21 '16 at 4:52
• @Ovi -- Yes, because the partial sums are just regular finite sums. But you still need to be careful that your partial sums are entire initial parts of the infinite series. You can't pick and choose which terms to include; all you can do is chop the infinite series off at some finite point. After that, you can apply any normal algebraic operations, because you just have a finite sum. – Mitchell Spector Oct 21 '16 at 4:56
• @ Mitchell Spectator Okay got it – Ovi Oct 21 '16 at 4:57
• And if you want to prove that the conditionally convergent series does converge, you can't cherry-pick which of the partial sums you're looking at (like it happens here), of course. – Henning Makholm Oct 21 '16 at 14:21
• @HenningMakholm Good point. – Mitchell Spector Oct 21 '16 at 15:37

To expand on Mitchell Spector's answer, let's generalize to the problem where every $b$th term is negated.

By the same argument, we get \begin{align} \sum_{n=1}^{bN}\frac{a_n}{n} &= \sum_{n=1}^{bN}\frac1{n}-2\sum_{n=1}^N \frac1{bn} \\&=\sum_{n=1}^{bN}\frac1{n}-\frac2{b}\sum_{n=1}^N \frac1{n} \\&\ge\sum_{n=1}^{bN}\frac1{n}-\frac2{b}\sum_{n=1}^{bN} \frac1{n} \\&=\left(1-\frac2{b}\right)\sum_{n=1}^{bN}\frac1{n} \end{align}

This is nice, since we can see that the divergence proof holds for $b \geq 3$, but fails for $b=2$. This is not strange, since the series for $b=2$ is known to converge to $\log 2$.

This proof isn't valid because of the Riemann series theorem; the original series could still converge even though some rearrangement of it diverges. It's important to focus on the partial sums.

You can argue along the lines that the series $$\sum_{k=0}^{\infty} \frac{1}{8k+1} = 1 + \frac{1}{9} + \frac{1}{17} + \cdots$$ already diverges by comparing its partial sums to the partial sums of the harmonic series $\frac{1}{8} \Big( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \Big).$

Note that \begin{align} &{\tiny\sum_{k=0}^\infty}{\tiny\left(\frac1{8k+1}+\frac1{8k+2}+\frac1{8k+3}+\frac1{8k+4}+\frac1{8k+5}+\frac1{8k+6}+\frac1{8k+7}-\frac1{8k+8}\right)}\\ &={\tiny\sum_{k=0}^\infty}{\tiny\left(\frac1{8k+1}+\frac1{8k+2}+\frac1{8k+3}+\frac1{8k+4}+\frac1{8k+5}+\frac1{8k+6}\right)+{\tiny\sum_{k=0}^\infty}\left(\frac1{8k+7}-\frac1{8k+8}\right)}\\ \end{align} For the left hand sum, we have \begin{align} &{\small\sum_{k=0}^\infty\left(\frac1{8k+1}+\frac1{8k+2}+\frac1{8k+3}+\frac1{8k+4}+\frac1{8k+5}+\frac1{8k+6}\right)}\\ &\ge{\small\frac68\sum_{k=0}^\infty\frac1{k+1}} \end{align} which diverges.

For the right hand sum, we have \begin{align} &{\small\sum_{k=0}^\infty\left(\frac1{8k+7}-\frac1{8k+8}\right)}\\ &\le{\small\frac1{56}+\frac18\sum_{k=1}^\infty\left(\frac1{8k}-\frac1{8k+8}\right)}\\ &=\frac{15}{448} \end{align} A divergent sum plus a convergent sum diverges.

• How did you evaluate $\dfrac 18 \sum_{k=1}^{\infty} \left( \dfrac {1}{8k}-\dfrac {1}{8k+8} \right) = \dfrac {1}{64} \sum_{k=1}^{\infty} \dfrac {1}{k(k+1)}$? I guess its not necessary to calculate the sum because we know it converges anyway, but I'm curious how you got a rational answer, knowing that $\sum_{k=1}^{\infty} \dfrac {1}{k^2} = \dfrac {\pi^2}{6}$ – Ovi Oct 24 '16 at 15:19
• \begin{align} \sum_{k=1}^\infty\frac1{k(k+1)} &=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+1}\right)\\ &=\lim_{n\to\infty}\left(\sum_{k=1}^n\frac1k-\sum_{k=2}^{n+1}\frac1k\right)\\ &=\lim_{n\to\infty}\left(1-\frac1{n+1}\right)\\ &=1 \end{align} which is a Telescoping Sum. – robjohn Oct 24 '16 at 15:50


# Help Evaluating $\lim_{x \to \infty} \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}$

Does anyone know how to evaluate the following limit? $$\lim_{x \to \infty} \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}$$ The answer is $\frac{1}{2}$, but I want to see a step by step solution if possible.

Multiply by the conjugate, then reduce: \begin{align*} \lim_{x\to\infty} \left(\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right) &= \lim_{x\to\infty} \left(\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \right) \cdot \frac{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} \\ &= \lim_{x\to\infty} \frac{(x + \sqrt{x + \sqrt{x}}) - x}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} \\ &= \lim_{x\to\infty} \frac{\sqrt{x + \sqrt{x}}}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} \cdot \frac{\dfrac{1}{\sqrt{x}}}{\dfrac{1}{\sqrt{x}}} \\ &= \frac{\sqrt{\lim\limits_{x\to\infty}\dfrac{1}{\sqrt{x}} + 1}}{\sqrt{1 + \lim\limits_{x\to\infty} \dfrac{\sqrt{x + \sqrt{x}}}{x}} + 1} \\ &= \frac{\sqrt{0 + 1}}{\sqrt{1 + 0} + 1} \\ &= \frac{1}{2} \end{align*}

• I like your answer! :) – dmk Jun 15 '14 at 1:46
• thanks i like ur answer to – user157128 Jun 15 '14 at 1:54
• as you did in step 3 to 4 when multiplied by 1/v5 ??? – user157128 Jun 15 '14 at 2:03

$$\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}=\frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}=\frac{\sqrt{1+\frac{1}{\sqrt{x}}}}{\sqrt{1+\sqrt{\frac{1}{x}+\frac{1}{x^{\frac{3}{2}}}}}+1}$$

I got the above by first multiplying by the conjugate to $\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}$ then dividing both the numerator and denominator by $\frac{1}{\sqrt{x}}$. Now take $x$ to infinity.

• the answer is 1/2, but I tried this and rationalize, but did not reach 0.5 – user157128 Jun 15 '14 at 1:38
• @user157128 what did you get? In the top I get $\sqrt{1 + 0} - 0$. What do you get in the bottom? – DanZimm Jun 15 '14 at 1:40

\begin{align} \ \lim_{x\rightarrow\infty}\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} &= \lim_{x\rightarrow\infty} \left( \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} \right) \cdot \frac{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}\\ \ &= \lim_{x\rightarrow\infty} \frac{x + \sqrt{x+\sqrt{x}}- x}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}\\ \ &= \lim_{x\rightarrow\infty} \frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}} \\ \ &= \lim_{x\rightarrow\infty} \frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}} \cdot \frac{1/\sqrt{x}}{1/\sqrt{x}} \\ \ &= \lim_{x\rightarrow\infty} \frac{\sqrt{1+\frac{1}{\sqrt{x}}}}{\sqrt{1+\frac{1}{x}+\frac{1}{x\sqrt{x}}}+1} \\ \ &= \frac{\sqrt{1}}{\sqrt{1}+1} \\ \ &= \frac{1}{2} \\ \end{align}

Using Taylor expansion:

$$\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}$$

$$\sqrt{x + \sqrt{x}+\dfrac{1}{2} + O(\frac{1}{\sqrt{x}})} - \sqrt{x}$$

$$\sqrt{x}+\dfrac{1}{2\sqrt{x}}(\sqrt{x}+\dfrac{1}{2}+ O(\frac{1}{\sqrt{x}})) + O(\frac{1}{\sqrt{x}})- \sqrt{x}$$

$$\dfrac{1}{2} + \dfrac{1}{4\sqrt{x}}$$

Which in the limit case tends to ${\dfrac{1}{2}}$.

The expansion I used was $(a+b)^{1/2} = \sqrt{a}+\frac{b}{2\sqrt{a}} + \cdots$

• It's not clear from what you've written why you get to ignore the error terms in your expansion. – Cheerful Parsnip Jun 15 '14 at 1:50
• @user157107 this seems like a fairly interesting way to go about it... and similar to what I was considering doing. Could you possibly add more details to justify your approach? – Squirtle Jun 15 '14 at 2:13


# Find the nonzero vectors $u,v,w$ that are perpendicular to the vector $(1,1,1,1)$ and to each other

Find the nonzero vectors $$u,v,w$$ that are perpendicular to the vector $$(1,1,1,1)$$ and to each other.

If I follow algebra, then I get complicated results to solve it as follows:

Let $$u=(u_1,u_2,u_3,u_4), \ v=(v_1,v_2,v_3,v_4) , \ w=(w_1,w_2,w_3,w_4)$$

Then, $$u \cdot (1,1,1,1)=v \cdot (1,1,1,1)=w \cdot (1,1,1,1)=0$$

Also $$u \cdot v=w \cdot u=v \cdot w=0$$.

These gives us

$$u_1+u_2+u_3+u_4=0, \\ v_1+v_2+v_3+v_4=0 , \\ w_1+w_2+w_3+w_4=0, \\ u_1v_1+u_2v_2+u_3v_3+u_4v_4=0, \\ u_1w_1+u_2w_2+u_3v_3+u_4v_4=0, \\ v_1w_1+v_2w_2+v_3w_3+v_4w_4=0.$$

But how to solve for $$u_i, v_i,w_i, \ i=1,2,3,4$$ from here?

Does there exit any other easy method?

Help me out

• Apply Gram-Schmidt to the basis $$((1, 1, 1, 1), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)),$$ or any other basis beginning with $(1, 1, 1, 1)$. – Theo Bendit Sep 27 '18 at 2:37
• Is not in that case the solution will be particular , I mean there can other vectors also which can not be conclude using Gram-schmidt method. – M. A. SARKAR Sep 27 '18 at 2:44
• Yes, it will be particular. If you range over all such bases, then you will obtain all orthonormal bases beginning with $\left(\frac12,\frac12,\frac12,\frac12\right)$, though not uniquely. – Theo Bendit Sep 27 '18 at 2:47

as columns $$\left( \begin{array}{rrrr} 1&-1&-1&-1 \\ 1& 1&-1&-1 \\ 1&0 &2&-1 \\ 1&0&0&3 \end{array} \right)$$ Pattern, done correctly, works in any dimension

$$\left( \begin{array}{rrrrrrrrrr} 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 0 & 2 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 3 & -1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 0 & 4 & -1 & -1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 0 & 0 & 5 & -1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 6 & -1 & -1 & -1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & -1 & -1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & -1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 \end{array} \right).$$

• That's great! =) – Siong Thye Goh Sep 27 '18 at 3:02
• Excellent method – M. A. SARKAR Sep 27 '18 at 11:13

Consider the Hadamard matrix of which we know that the columns form an orthogonal basis of $$\mathbb{R}^4$$.

$$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1\end{bmatrix}$$

The other columns would give you a solution.

Alternatively, use Gram-Schmidt process.

• But if we consider the Hadamard matrix , then the solution will be particular. There may be other vectors which are perpendicular to $(1,1,1,1)$ except the last 3 column vectors in the Hadamard vectors and its multiples . – M. A. SARKAR Sep 27 '18 at 2:42
• So if we use Gram-Schmidt method then we need a basis – M. A. SARKAR Sep 27 '18 at 2:43
• Theo has given you a basis right? Note that answer is not unique. If you want to describe the set, you have already done so in your post. – Siong Thye Goh Sep 27 '18 at 2:48
• There is a pattern that easily adapts to any dimension... – Will Jagy Sep 27 '18 at 3:01


# Is it necessary to write limits for a substituted integral?

To solve the following integral, one can use u-substitution: $$\int_2^3 \frac{9}{\sqrt[4]{x-2}} \,dx,$$ With $$u = \sqrt[4]{x-2}$$, our bounds become 0 and 1 respectively. Thus, we end up with the following: $$36\int_0^1{u^2} \,du$$ In the first case, the lower bound is a vertical asymptote so we would have to use limits to find the answer. However, in the second case there's no longer an asymptotal bound - would you still have to write the limits since the original function would've needed limits, or can you just solve this by plugging in the substituted bounds? I know the final answer will be the same either way, but I want to know if it can be considered correct to exclude the limits in the 2nd integral from a technical perspective since the function has been changed. Many thanks in advance!

## 3 Answers

Without too much simplification, the substitution you cite yields

$$\int_2^3\frac9{(x-2)^{1/4}}\,\mathrm dx=36\int_0^1\frac{u^3}u\,\mathrm du$$

which you certainly welcome to treat as an improper integral,

$$36\left(\frac13-\lim_{u\to0^+}\frac{u^3}3\right)$$

but since $$u=0$$ is a removable discontinuity and the limand reduces to $$u^2$$, you may as skip this treatment altogether.

• I'm confused about your last expression. It seems like you want to still have $\int$ present, but the $1-$ seems like you've already taken the integral, but the $\frac{u^3}u$ is from before you've taken the integral... – Teepeemm Feb 20 at 2:05
• Yes, it was wrong as written. I have edited it. – Martin Argerami Feb 20 at 3:51
• Thanks @MartinArgerami ! – user170231 Feb 20 at 14:55

There are certain conditions that must be met for a substitution to be "legal". In most circumstances these conditions are naturally met and so they are not emphasized; you have here one situation in which they are not.

For instance, one set of conditions is given in Anton, Anderson, and Bivens (Calculus, Early Transcendentals, 11th Edition), Theorem 5.9.1:

Theorem 5.9.1. If $$g'(x)$$ is continuous on $$[a,b]$$, and $$f$$ is continuous on an interval containing the values of $$g(x)$$ for $$a\leq x\leq b$$, then $$\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du.$$

Here you have $$g(x) = \sqrt[4]{x-2}$$, so $$g'(x) = \frac{1}{4}(x-2)^{-3/4}$$... which is not continuous on the interval $$[2,3]$$ that you are working on.

In fact, as you note, the initial integral is improper, which means you aren't really evaluating that integral: you are evaluting a limit, $$\lim_{h\to 2^+}\int_h^3 \frac{9}{\sqrt[4]{x-2}}\,dx.$$ The integral in the limit does satisfy the conditions of the theorem above, so you can make the substitution to get $$\lim_{h\to 2^+}\int_{\sqrt[4]{h-2}}^1 \frac{u^3}{u}\,du = \lim_{a\to 0^+}\int_a^1 u^2\,du,$$ and proceed from there.

• Thank you very much Arturo, your answer was very in-depth and provided a lot of reinforcement for what you were saying. I wish I could select multiple answers to be correct, I thought your inclusion of the cited textbook snippet was really nicely done too. I chose the other answer only because I found it easier to digest/understand at first glance, but after I properly read your comment I understood it more fully. Thank you for your quick and effective contribution :-) – Lord Kanelsnegle Feb 19 at 17:32

Suppose you must. Then we have $$9\lim_{a\to2^+}\int_a^3\frac1{\sqrt[4]{x-2}}\,dx.$$

Let $$u=x-2$$. Then we have $$9\lim_{a\to2^+}\int_{a-2}^1 u^{-1/4}\, du.$$

By the power rule, we have $$9\lim_{a\to2^+} \left.\frac43u^{3/4}\right]_{a-2}^1=9\left[\frac43(1-\lim_{a\to2^+}(a-2)^{3/4})\right]=9\left[\frac43(1-0)\right]=12.$$

Notice that it does not make a difference whether you use $$\lim_{a\to2^+}(a-2)$$ or $$0$$.

• Thanks for the quick and simple response! I found the answer I selected to be more aligned with the question, but I wanted to comment because I appreciate the time you took to write this. Thank you once again. – Lord Kanelsnegle Feb 19 at 17:29


# Expected Value of flipping through a book

Suppose that a book has $$N$$ pages, and we read through the book as follows. We start from page 0, and if we're on page $$i$$, we randomly flip to a page $$i + 1, i + 2, ..., N$$ with equal probability.

1. What is the expected value of number of flips we need to finish the book?

Intuition tells me that we can, on average, expect to halve the number of pages remaining. This yields $$\log_2(N)$$, but I'm having trouble formalizing it.

1. If $$N = 26$$, what is the probability we flip to page 13 at some point? Assume we start at page 0.

I let $$P_i$$ be the probability we land on page 13 eventually, starting from page $$i$$. Then, $$P_{13} = 1$$, and in general, $$P_{i} = \frac{1}{26 - i}\sum_{k = i + 1}^{13}P_k$$

Evaluating terms like $$P_{12}, P_{11}, P_{10}$$, I see that all of these values are $$\frac{1}{14}$$, including $$P_0$$. Is there a more intuitive reason for such a simple answer?

• Question 1. is unclear ... number of flips we need for what? To read every page in the book? To reach the last page in the book? What does it mean? – Ross Presser Aug 21 at 13:47
• edited for clarity, though it should be implicitly clear that it's often not possible to read every page in the book, and the expected value of flips to read every page is trivial. – user815048 Aug 21 at 20:13

Let's consider the equivalent problem in which we start at page $$n$$ and flip backwards through the book, going to each of the pages $$0, 1, ..., n - 1$$ with equal probability. Let $$E_n$$ be the expected number of flips. Then we have $$E_0 = 0$$ and

$$E_n = 1 + \frac{1}{n} \sum\limits_{i = 0}^{n - 1} E_i$$

Then in particular we have

$$$$\begin{split} E_{n + 1} &= 1 + \frac{1}{n + 1} \sum\limits_{i = 0}^{n} E_i \\ &= 1 + \frac{E_n}{n + 1} + \frac{n}{n + 1} \frac{1}{n} \sum\limits_{i = 0}^{n - 1} E_i \\ &= 1 + \frac{E_n}{n + 1} + \frac{n}{n + 1} (1 + \frac{1}{n} \sum\limits_{i = 0}^{n - 1} E_i) - \frac{n}{n + 1} \\ &= 1 - \frac{n}{n + 1} + \frac{1}{n + 1} E_n + \frac{n}{n + 1} E_n \\ &= \frac{1}{n + 1} + E_n \end{split}$$$$

whenever $$n \geq 1$$ (and the identity is easily verified when $$n = 0$$ as well).

Then by induction, we have $$E_n = \sum\limits_{j = 1}^n \frac{1}{j}$$, the $$n$$th harmonic number. This will be asymptotically very close to $$\log_e(n)$$.

Let $$P_n$$ be the expected number of flips in a book with $$n$$ pages. Then $$P_0=0,\ P_1=1$$ and $$P_n=1+\frac1n\sum_{k=0}^{n-1}P_k,\tag1$$ because we have to make one flip, and then we're equally likely to have any number of pages from $$0$$ to $$n-1$$ left to flip through.

We get \begin{align} P_1&=1\\ P_2&=\frac32\\ P_3&=\frac{11}6\\ P_4&=\frac{50}{24}\\ P_5&=\frac{174}{120} \end{align}

The denominators are obviously $$n!$$, so we look for the numerators in OEIS and find A000254, the unsigned Stirling numbers of the first kind.

OESI gives the recurrence $$a_{n+1}=(n+1)a_n+n!$$ for the unsigned Stirling numbers of the first kind, and dividing through by $$(n+1)!$$ we get $$P_{n+1}=P_n+\frac1{n+1}$$ which clearly gives $$P_n=\sum_{k=1}^n\frac1k=H_n,$$ the $$n$$th harmonic number. To complete the problem, we must show that the harmonic numbers satisfy the recurrence $$(1)$$.

Here is how I approached the first part of the problem. Consider a book with exactly $$n$$ pages. Let $$P_1$$ denote the first page you flipped to, and let $$X_n$$ represent the number of pages you flipped until you get to the last page. Note $$P_1$$ is uniformly distributed on the set $$\{1,...,n\}$$ and $$E(X_1)=1$$. Using the total law of expectation we get for $$n\geq2$$ that $$E(X_n)=\sum_{k=1}^{n}E(X_n|P_1=k)P(P_1=k)=\frac{1}{n}\sum_{k=1}^{n}E(X_n|P_1=k)$$
Notice $$E(X_n|P_1=k)=1+E(X_{n-k})$$ and so $$E(X_n)=\frac{1}{n}\sum_{k=1}^{n}\Big[1+E(X_{n-k})\Big]=1+\frac{E(X_0)+\dots+E(X_{n-1})}{n}$$ Replace $$n$$ with $$n+1$$ to get $$E(X_{n+1})=1+\frac{E(X_0)+\dots+E(X_{n})}{n+1}$$ Combining the previous two equations unveils the relation $$(n+1)(E(X_{n+1})-1)=(n+1)E(X_n)-n$$ which is equivalent to saying $$E(X_{n+1})=E(X_n)+\frac{1}{n+1}$$ So finally $$E(X_n)=\sum_{k=1}^{n}\frac{1}{k}$$


# Math Help - college prob + stat

1. ## college prob + stat

2 Questions:

1) There are 5 men and 5 women in a group. A committee is to be formed of 5 people. What is the probability that the group is all of the same sex (all men or all women)?

2) There are 10 people in an elevator in the basement. They can get off at any of 5 floors above. The probability that they get off at each floor is equal and each person leaving is independant of each other. What is the probability that 2 people get off at each floor?

2. Originally Posted by magicpunt
1) There are 5 men and 5 women in a group. A committee is to be formed of 5 people. What is the probability that the group is all of the same sex (all men or all women)?
$P(\mbox{men or woman})=P(\mbox{men})+P(\mbox{woman})$
Let us find,
$P(\mbox{men})$
There are 10 people and 5 are selected there are a total of $_{10}C_5=252$
From 5 men you take 5 men intotal the number of combinations is,
$_5C_5=1$
Thus,
$P(\mbox{men})=\frac{1}{252}$
Similarily ya have,
$P(\mbox{women})=\frac{1}{252}$
Thus, the probability is,
$\frac{1}{252}+\frac{1}{252}=\frac{2}{252}=\frac{1} {126}$

3. 2) There are 10 people in an elevator in the basement. They can get off at any of 5 floors above. The probability that they get off at each floor is equal and each person leaving is independant of each other. What is the probability that 2 people get off at each floor?
This is similar to the 'boxes and balls' problems.

There are $5^{10}=9765625$ ways to distribute the people to the 5 floors.

There are $\frac{5!}{2!}=60$ ways that no two people get off on the same floor.

EDIT:
Yes, I misinterpreted the problem. Cerebral flatulence. Soroban's method is solid.
My second part should be $\frac{10!}{2^{5}}$

Therefore, $\frac{5^{10}}{\frac{10!}{2^{5}}}$

This is a version of the classic 'assigning diplomats' problem.

For instance:

How many ways can 10 different diplomats be assigned to 5 different countries if 2 diplomats must be assigned to each country?.

See the similarity with your problem?. Yours is just elevators and people, instead.

This can be done in $\frac{10!}{(2!)^{5}}=113,400$ ways.

The method I used is just another variation on this problem. Keep it, you may need it.

4. Originally Posted by galactus
This is similar to the 'boxes and balls' problems.

There are $5^{10}=9765625$ ways to distribute the people to the 5 floors.

There are $\frac{5!}{2!}=60$ ways that no two people get off on the same floor.
I think you might've misunderstood the problem

The question is: what is P(Exactly 2 people get off at each floor)

5. Hello, magicpunt!

Here's my approach to #2 . . .

2) There are 10 people in an elevator in the basement.
They can get off at any of 5 floors above.
The probability that they get off at each floor is equal
and each person leaving is independant of each other.
What is the probability that 2 people get off at each floor?

Each of the ten people has a choice of five floors.
. . There are: $5^{10}$ways that they can leave the elevator.

If exactly two people get off at each floor,
. . the number of ways is: $\begin{pmatrix}10\\2,2,2,2,2\end{pmatrix} = 113,400$

Therefore, the probabilty is: . $\frac{113,400}{5^{10}}\:=\:\frac{4,536}{390,625}$

6. Originally Posted by magicpunt
2 Questions:

1) There are 5 men and 5 women in a group. A committee is to be formed of 5 people. What is the probability that the group is all of the same sex (all men or all women)?
Another approach to #1
Find the total no. of ways of selecting 5 out of 10= $\frac{10!}{5!5!}=C_5^{10}$
No. of ways in which all are men=1
No. of ways in which all are women=1
Required probability= $\frac{1+1}{C_5^{10}}=\frac{1}{126}$

Malay