Question

A sphere and a cylinder have the same radius and height. The volume of the cylinder is 30 m

Answer 1

Answer:

The volume of the sphere is 20 m

Step-by-step explanation:

Given

Solid Shapes: Cylinder and Sphere

Volume of the cylinder $= 30m^3$

Required

Volume of the sphere

First, we need to calculate the radius of the cylinder

The formula goes thus

$V = \pi r^2h$

By substituting 30 for V, we have

$30 = \pi r^2h$

Divide through by h

$\frac{30}{h} = \frac{\pi r^2h}{h}$

$\frac{30}{h} = \pi r^2$

Calculating the volume of the sphere

The formula goes thus

$V = \frac{4}{3}\pi r^3$

Expand Expression

$V = \frac{4}{3} * \pi r^2 * r$

Substitute $\frac{30}{h} = \pi r^2$

$V = \frac{4}{3} * \frac{30}{h} * r$

Simplify Expression

$V = \frac{4}{1} * \frac{10}{h} * r$

$V = \frac{40}{h} * r$

Given that the height of the cylinder and the sphere are equal

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

Mathematically; This is represented by

h = D

Where D represents diameter of the sphere

Recall that D = 2r (2 * radius)

Substitute 2r for D

h = 2r

Substitute h = 2r in $V = \frac{40}{h} * r$; This gives

$V = \frac{40}{2r} * r$

$V = \frac{40r}{2r}$

$V = 20$

Hence, the volume of the sphere is 20 m