Find the volume of the solid whose base is a circle of radius r and whose cross-sections are squares perpendicular to a diameter.

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vaaruni's profile pic

vaaruni | High School Teacher | (Level 1) Salutatorian

Posted on

The given solid should be a cylinder because only a cylinder when,

we cut/sliced crossections perpendicular to the diameter, each cross-

section  will be a rectangle and if the height of the cylinder is equal to

its diameter the the cross-section is a square. Hence, volume of the

given solid = volume of the cylinder of height equal to its diameter(2r)

Volume of the cylinder = pi * r^2 * h = pi*(r)^2 *(2r) = 2pi*(r)^3    

Volume of the given solid = 2*pi*(r)^3  <----  Answer     

quantatanu's profile pic

quantatanu | Student, Undergraduate | (Level 1) Valedictorian

Posted on

It seems that its a cillynder with length L=2*r, 

because only then if you slice it in a direction perpendicular to one of the diameter of the base-circle you will get squares ! As we are getting squares and not rectangles in general that means the length of the cillynder must be equal to the diameter of the circle because for it to be square length must be equal to breadth (base circle diameter) so 2*r length cillynder.

So volume = area of base x length of cillynder = (pi) x (r^2) x (2 r)

                                                      volume = 2 pi r^3

No it cannot be a cilynder, because when one moves off the center of the cilynder and cut slices perpendicular to the base one doesn't get squares but rectangles ! 

Consider the centralsquare that is the square which stands over the center of the circle, then it must have a length of diameter of the circle = 2*r, now if we move "x" distance away from the centre then the length must dicrease and its now a "chord" so the "chord length" at a distance "x" away from center is,

L(x) = 2 * Sqrt[r^2 - x^2], we take this square slice of thickness "dx" and integrate from center to the edge and then double it and we get the volume.

volume of the slice (dV) = [L(x)]^2 dx

                                      = 4 * (r^2 - x^2) * dx

Volume of the whole solid = 2* Integral [4 * (r^2 - x^2) * dx] from "0" to "r"

                                         = 8* [r^2 x - x^3 /3] from 0  to  r

                                         = 8* [r^3 - r^3 /3]

                                         = (16/3) r^3  

najm1947's profile pic

najm1947 | Elementary School Teacher | (Level 1) Valedictorian

Posted on

Answer 1 & 2 are correct because it says:

whose cross-sections are squares perpendicular to a diameter

and all cross-sections perpendicular to the diameter of a cylinder of length equal to diameter =2r, where r is the radius, will make a square. All the cross-sections will pass through the center whichever diameter we may select.

quantatanu's profile pic

quantatanu | Student, Undergraduate | (Level 1) Valedictorian

Posted on

It seems that its a cillynder with length L=2*r, 

because only then if you slice it in a direction perpendicular to one of the diameter of the base-circle you will get squares ! As we are getting squares and not rectangles in general that means the length of the cillynder must be equal to the diameter of the circle because for it to be square length must be equal to breadth (base circle diameter) so 2*r length cillynder.

So volume = area of base x length of cillynder = (pi) x (r^2) x (2 r)

                                                      volume = 2 pi r^3

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