To solve this problem, let's follow these steps:
1. Calculate the radius of Sphere A:
- Given the diameter of Sphere A is 12.
- The radius is half of the diameter.
- Therefore, the radius of Sphere A is [tex]\( \frac{12}{2} = 6 \)[/tex] units.
2. Determine the radius of Sphere B after dilation:
- Sphere A is dilated by a scale factor of [tex]\( \frac{1}{2} \)[/tex].
- The radius of Sphere B is [tex]\( 6 \times \frac{1}{2} = 3 \)[/tex] units.
3. Compute the volume of Sphere A:
- The volume [tex]\( V \)[/tex] of a sphere is given by the formula [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex], where [tex]\( r \)[/tex] is the radius.
- For Sphere A with radius 6, the volume is:
[tex]\[
V_A = \frac{4}{3} \pi (6)^3 \approx 904.78 \text{ cubic units}
\][/tex]
4. Compute the volume of Sphere B:
- For Sphere B with radius 3, the volume is:
[tex]\[
V_B = \frac{4}{3} \pi (3)^3 \approx 113.10 \text{ cubic units}
\][/tex]
5. Find the ratio of the volumes:
- The ratio of the volume of Sphere A to Sphere B is:
[tex]\[
\text{Ratio} = \frac{V_A}{V_B} = \frac{904.78}{113.10} = 8
\][/tex]
So, the ratio of the volume of Sphere A to Sphere B is [tex]\( 8:1 \)[/tex].
