Question

1. in which of these cases will a spheres volume increase by more than 200π cm³? select two correct answers.

a. when its radius increases from 2 cm to 4 cm
b. when its radius increases from 3 cm to 5 cm
c. when its radius increases from 5 cm to 6 cm
d. when its radius increases from 6 cm to 7 cm

Explanation:

Step1: Recall the volume formula of a sphere

The volume formula of a sphere is \(V=\frac{4}{3}\pi r^{3}\). The increase in volume \(\Delta V=\frac{4}{3}\pi(r_{2}^{3}-r_{1}^{3})\).

Step2: Calculate the volume increase for each option

• Option A:

\(r_{1} = 2\), \(r_{2}=4\)
\(\Delta V=\frac{4}{3}\pi(4^{3}-2^{3})=\frac{4}{3}\pi(64 - 8)=\frac{4}{3}\pi\times56=\frac{224}{3}\pi\approx74.67\pi<200\pi\)

• Option B:

\(r_{1}=3\), \(r_{2} = 5\)
\(\Delta V=\frac{4}{3}\pi(5^{3}-3^{3})=\frac{4}{3}\pi(125 - 27)=\frac{4}{3}\pi\times98=\frac{392}{3}\pi\approx130.67\pi<200\pi\)

• Option C:

\(r_{1}=5\), \(r_{2}=6\)
\(\Delta V=\frac{4}{3}\pi(6^{3}-5^{3})=\frac{4}{3}\pi(216 - 125)=\frac{4}{3}\pi\times91=\frac{364}{3}\pi\approx121.33\pi<200\pi\)

• Option D:

\(r_{1}=6\), \(r_{2}=7\)
\(\Delta V=\frac{4}{3}\pi(7^{3}-6^{3})=\frac{4}{3}\pi(343-216)=\frac{4}{3}\pi\times127=\frac{508}{3}\pi\approx169.33\pi<200\pi\)

Answer:

None of the options A - D have a volume increase more than \(200\pi\mathrm{cm}^{3}\). There might be a mistake in the problem statement or options provided. If we assume the formula application is correct, based on the calculations above, no correct options among A - D.