Question

A sphere and a cylinder have the same radius and height. The volume of the cylinder is 27 pi feet cubed.

Which equation gives the volume of the sphere?
V = two-thirds (27 pi)
V = four-thirds (27)
V = two-thirds (27)
V = four-third (27 pi)

Answer 1

C. V = two-thirds (27)

Explanation:

Given

Solid Shapes: Cylinder and Sphere

Volume of Cylinder = 27π ft³

Required

Volume of the sphere.

From the question,

We have that

1. The volume of the sphere is the same as the volume of the cylinder

2. The height of the sphere is the same as the height of the cylinder.

From (2) above;

This means that the height of the cylinder equals the diameter of the sphere.

Let h represent the height of the sphere and d represent the diameter of sphere.

Mathematical, d = h

Recall that radius, r =
`r = (d)/(2)`

Substitute h for d in the above expression

`r = (h)/(2)`. ----- (take note of this)

Calculating the volume of a cylinder.

V = πr²h

Recall that V = 27; This gives us

27 = πr²h

Divide both sides by h

`\pi r^2 = (27)/(h)`

-------------------

Calculating the volume of a sphere

`V = (4)/(3)\pi r^3`

Expand the above expression

`V = (4)/(3)\pi r^2 * r`

Substitute
`\pi r^2 = (27)/(h)`

`V = (4)/(3) * (27)/(h) * r`

Recall that
`r = (h)/(2)`

So,

`V = (4)/(3) * (27)/(h) * (h)/(2)`

`V = (4)/(3) * (27)/(2)`

`V = (2)/(3) * 27`

`V = (2)/(3) (27)`

V = two-third (27)