Question

1. how does the volume of a sphere with a radius of 2 meters compare to the volume of a sphere with a radius of 4 meters?

Explanation:

Step1: Recall the volume formula of a sphere

The volume \( V \) of a sphere is given by the formula \( V = \frac{4}{3}\pi r^{3} \), where \( r \) is the radius of the sphere.

Step2: Calculate the volume of the sphere with radius \( r_1 = 2 \) meters

Substitute \( r = 2 \) into the formula:
\( V_1=\frac{4}{3}\pi(2)^{3}=\frac{4}{3}\pi\times8=\frac{32}{3}\pi \) cubic meters.

Step3: Calculate the volume of the sphere with radius \( r_2 = 4 \) meters

Substitute \( r = 4 \) into the formula:
\( V_2=\frac{4}{3}\pi(4)^{3}=\frac{4}{3}\pi\times64=\frac{256}{3}\pi \) cubic meters.

Step4: Find the ratio of \( V_1 \) to \( V_2 \)

To compare the two volumes, we find the ratio \( \frac{V_1}{V_2} \):
\( \frac{V_1}{V_2}=\frac{\frac{32}{3}\pi}{\frac{256}{3}\pi} \)
The \( \frac{1}{3}\pi \) terms cancel out, leaving \( \frac{32}{256}=\frac{1}{8} \).

Answer:

The volume of the sphere with a radius of 2 meters is \( \frac{1}{8} \) of the volume of the sphere with a radius of 4 meters (or the volume of the sphere with radius 4 meters is 8 times that of the sphere with radius 2 meters).