Let's break down the problem step-by-step.
1. Finding the radii of the spheres:
Given surface areas:
- Surface area of the larger sphere: [tex]\( 100 \pi \)[/tex] units²
- Surface area of the smaller sphere: [tex]\( 36 \pi \)[/tex] units²
The formula for the surface area of a sphere is:
[tex]\[
4\pi r^2
\][/tex]
From this, we can solve for the radius [tex]\( r \)[/tex] of each sphere.
For the larger sphere:
[tex]\[
4\pi r_{\text{large}}^2 = 100 \pi
\][/tex]
[tex]\[
r_{\text{large}}^2 = \frac{100 \pi}{4 \pi} = 25
\][/tex]
[tex]\[
r_{\text{large}} = \sqrt{25} = 5 \text{ units}
\][/tex]
For the smaller sphere:
[tex]\[
4\pi r_{\text{small}}^2 = 36 \pi
\][/tex]
[tex]\[
r_{\text{small}}^2 = \frac{36 \pi}{4 \pi} = 9
\][/tex]
[tex]\[
r_{\text{small}} = \sqrt{9} = 3 \text{ units}
\][/tex]
2. Determining the scale factor:
The scale factor is the ratio of the radii of the larger sphere to the smaller sphere.
[tex]\[
\text{Scale factor} = \frac{r_{\text{large}}}{r_{\text{small}}} = \frac{5}{3} \approx 1.667
\][/tex]
3. Finding the volume of the smaller sphere:
Given the volume of the larger sphere:
[tex]\[
\frac{500}{3} \pi \text{ units}^3
\][/tex]
We know the formula for the volume of a sphere is:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
For the smaller sphere, using its radius [tex]\( r_{\text{small}} = 3 \)[/tex]:
[tex]\[
V_{\text{small}} = \frac{4}{3} \pi r_{\text{small}}^3
\][/tex]
[tex]\[
V_{\text{small}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ units}^3
\][/tex]
So, summarizing the results:
- Radius of the larger sphere: [tex]\( 5 \)[/tex] units
- Radius of the smaller sphere: [tex]\( 3 \)[/tex] units
- Scale factor: [tex]\( 1.667 \)[/tex]
- Volume of the smaller sphere: [tex]\( \approx 113.097 \)[/tex] units³
These are the complete answers required.
