Question

if the radius of a sphere increases from 4 cm to 8 cm, by what factor does the volume increase? a. 6 b. 2 c. 8 d. 4

Explanation:

Step1: Recall the volume formula of a sphere

The volume formula of a sphere is \(V = \frac{4}{3}\pi r^{3}\).

Step2: Calculate the initial volume \(V_1\)

When \(r_1 = 4\space cm\), \(V_1=\frac{4}{3}\pi(4)^{3}=\frac{4}{3}\pi\times64\).

Step3: Calculate the final volume \(V_2\)

When \(r_2 = 8\space cm\), \(V_2=\frac{4}{3}\pi(8)^{3}=\frac{4}{3}\pi\times512\).

Step4: Find the ratio \(\frac{V_2}{V_1}\)

\(\frac{V_2}{V_1}=\frac{\frac{4}{3}\pi\times512}{\frac{4}{3}\pi\times64}\). Cancel out \(\frac{4}{3}\pi\), we get \(\frac{512}{64}=8\).

Answer:

C. 8