Question

1. what happens to the volume of a sphere if its radius is halved?

a. the volume stays the same.
b. the volume is reduced to one - fourth.
c. the volume is halved.
d. the volume is reduced to one - eighth.

Explanation:

Step1: Recall the volume formula of a sphere

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

Step2: Calculate the new volume when the radius is halved

Let the original radius be \(r\), and the new radius be \(r'=\frac{r}{2}\). The new volume \(V'=\frac{4}{3}\pi(r')^{3}\). Substitute \(r' = \frac{r}{2}\) into the formula: \(V'=\frac{4}{3}\pi(\frac{r}{2})^{3}=\frac{4}{3}\pi\frac{r^{3}}{8}=\frac{1}{8}\times\frac{4}{3}\pi r^{3}\). Since \(V = \frac{4}{3}\pi r^{3}\), then \(V'=\frac{1}{8}V\).

Answer:

d. The volume is reduced to one - eighth.