Let's solve this step-by-step:
1. Find the radius of the larger sphere:
Given the surface area [tex]\(A\)[/tex] of the larger sphere is [tex]\(100 \pi\)[/tex] units[tex]\(^2\)[/tex]:
The formula for the surface area of a sphere is:
[tex]\[
A = 4 \pi r^2
\][/tex]
Solving for [tex]\(r\)[/tex]:
[tex]\[
r = \sqrt{\frac{A}{4 \pi}}
\][/tex]
Substituting [tex]\(A = 100 \pi\)[/tex]:
[tex]\[
r = \sqrt{\frac{100 \pi}{4 \pi}} = \sqrt{\frac{100}{4}} = \sqrt{25} = 5 \text{ units}
\][/tex]
So, the radius of the larger sphere is 5 units.
2. Find the radius of the smaller sphere:
Given the surface area [tex]\(A\)[/tex] of the smaller sphere is [tex]\(36 \pi\)[/tex] units[tex]\(^2\)[/tex]:
Again using the surface area formula:
[tex]\[
r = \sqrt{\frac{A}{4 \pi}}
\][/tex]
Substituting [tex]\(A = 36 \pi\)[/tex]:
[tex]\[
r = \sqrt{\frac{36 \pi}{4 \pi}} = \sqrt{\frac{36}{4}} = \sqrt{9} = 3 \text{ units}
\][/tex]
So, the radius of the smaller sphere is 3 units.
3. Determine the scale factor:
The scale factor is the ratio of the radii of the two spheres:
[tex]\[
\text{Scale factor} = \frac{\text{Radius of smaller sphere}}{\text{Radius of larger sphere}} = \frac{3}{5} = 0.6
\][/tex]
4. Find the volume of the smaller sphere:
The volume [tex]\(V\)[/tex] of a sphere is given by:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
Substituting [tex]\(r = 3\)[/tex] units:
[tex]\[
V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \cdot 27 = \frac{108}{3} \pi = 36 \pi \text{ units}^3
\][/tex]
Therefore, the volume of the smaller sphere is approximately 113.09733552923254 [tex]\(\pi\)[/tex] units[tex]\(^3\)[/tex].
To summarize, the required measures are:
- Scale factor: 0.6
- Radius of the smaller sphere: 3 units
- Radius of the larger sphere: 5 units
- Volume of the smaller sphere: 113.09733552923254 [tex]\(\pi\)[/tex] units[tex]\( ^3 \)[/tex]
