To determine the ratio of the volume of Sphere A to the volume of Sphere B after a dilation by a scale factor of 3, follow these steps:
1. Find the radius of Sphere A:
- Given that the diameter of Sphere A is 2, the radius (which is half the diameter) is:
[tex]\[
\text{radius\_A} = \frac{\text{diameter\_A}}{2} = \frac{2}{2} = 1
\][/tex]
2. Calculate the radius of Sphere B:
- Since the dilation scale factor is 3, the radius of Sphere B is:
[tex]\[
\text{radius\_B} = \text{radius\_A} \times \text{scale\_factor} = 1 \times 3 = 3
\][/tex]
3. Calculate 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]
- Substituting the radius of Sphere A:
[tex]\[
\text{volume\_A} = \frac{4}{3} \pi (1^3) = \frac{4}{3} \pi (1) = \frac{4}{3} \pi \approx 4.1887902047863905
\][/tex]
4. Calculate the volume of Sphere B:
- Using the same volume formula for Sphere B:
[tex]\[
\text{volume\_B} = \frac{4}{3} \pi (3^3) = \frac{4}{3} \pi (27) = 36 \pi \approx 113.09733552923254
\][/tex]
5. Determine the ratio of the volumes:
- The ratio of the volume of Sphere A to the volume of Sphere B is:
[tex]\[
\text{volume\_ratio} = \frac{\text{volume\_A}}{\text{volume\_B}} = \frac{4.1887902047863905}{113.09733552923254} \approx 0.03703703703703704
\][/tex]
Therefore, the ratio of the volume of Sphere A to the volume of Sphere B is approximately [tex]\(0.037\)[/tex].
