Question

Sphere A has a diameter of 12 and is dilated by a scale factor of one half to create sphere B. What is the ratio of the volume of sphere A to sphere B?

2:1
8:1
12:6
144:36

Answer 1

To find the ratio of the volumes of sphere A to sphere B after dilation, we follow these steps:

1. **Calculate the radius of Sphere A:**
The diameter of Sphere A is 12. Therefore, the radius \( r_A \) is:
\[
r_A = \frac{12}{2} = 6
\]

2. **Calculate the radius of Sphere B after dilation:**
Sphere B is created by dilating Sphere A by a scale factor of \( \frac{1}{2} \). Thus, the radius \( r_B \) of Sphere B is:
\[
r_B = r_A \times \frac{1}{2} = 6 \times \frac{1}{2} = 3
\]

3. **Use the formula for the volume of a sphere:**
The volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]

4. **Calculate the volume of Sphere A:**
Substitute \( r_A = 6 \):
\[
V_A = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \times 216 = \frac{864}{3} \pi = 288 \pi
\]

5. **Calculate the volume of Sphere B:**
Substitute \( r_B = 3 \):
\[
V_B = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = \frac{108}{3} \pi = 36 \pi
\]

6. **Find the ratio of the volumes:**
The ratio of the volume of Sphere A to Sphere B is:
\[
\text{Ratio} = \frac{V_A}{V_B} = \frac{288 \pi}{36 \pi} = \frac{288}{36} = 8
\]

Thus, the ratio of the volume of Sphere A to Sphere B is:
\[
\boxed{8:1}
\]