Question

If the radius of a sphere is tripled, what is the ratio of the volume of the original sphere to that of the new sphere?

• A. 9 : 1
• B. 27 : 1
• C. 1 : 27
• D. 1 : 9

Answer 1

Answer:

C. 1 : 27

Let's solve this step by step.

The volume of a sphere is given by the formula:

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

Let the original radius be (r)

Then the original volume $(V_1)$ is:
$V_1 = \frac{4}{3} \pi r^3$

If the radius is tripled

New radius = (3r)

Then the new volume $(V_2)$ is:

$V_2 = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi (27 r^3) = 27 \cdot \frac{4}{3} \pi r^3$

$So (V_2 = 27 V_1).$

Ratio of original volume to new volume

$\text{Ratio} = \frac{V_1}{V_2} = \frac{V_1}{27 V_1} = \frac{1}{27}$

Answer:

$\boxed{1 : 27}$