Let's go through the problem step-by-step to find all the required measures.
1. Scale Factor:
The surface areas of the two spheres are given as [tex]\( 100\pi \)[/tex] square units and [tex]\( 36\pi \)[/tex] square units. To find the scale factor, we use the ratio of the surface areas and take the square root of that ratio because the surface area of a sphere is proportional to the square of its radius.
[tex]\[
\text{Scale Factor} = \sqrt{\frac{\text{Surface Area of Larger Sphere}}{\text{Surface Area of Smaller Sphere}}} = \sqrt{\frac{100\pi}{36\pi}} = \sqrt{\frac{100}{36}} = \sqrt{\frac{25}{9}} = \frac{5}{3} = 1.6667
\][/tex]
Thus, the scale factor is [tex]\( 1.6667 \)[/tex].
2. Radius of the Smaller Sphere:
The surface area of the sphere is given by the formula [tex]\( 4\pi r^2 \)[/tex]. Using the surface area of the smaller sphere:
[tex]\[
4\pi r_{\text{small}}^2 = 36\pi
\][/tex]
Dividing both sides by [tex]\( 4\pi \)[/tex],
[tex]\[
r_{\text{small}}^2 = 9 \implies r_{\text{small}} = \sqrt{9} = 3
\][/tex]
So, the radius of the smaller sphere is [tex]\( 3 \)[/tex] units.
3. Radius of the Larger Sphere:
Similarly, using the surface area of the larger sphere:
[tex]\[
4\pi r_{\text{large}}^2 = 100\pi
\][/tex]
Dividing both sides by [tex]\( 4\pi \)[/tex],
[tex]\[
r_{\text{large}}^2 = 25 \implies r_{\text{large}} = \sqrt{25} = 5
\][/tex]
So, the radius of the larger sphere is [tex]\( 5 \)[/tex] units.
4. Volume of the Smaller Sphere:
The volume of a sphere is given by the formula [tex]\( \frac{4}{3}\pi r^3 \)[/tex]. We can use the ratio of the cube of the radii to find the volume of the smaller sphere. Given that the volume of the larger sphere is [tex]\( \frac{500}{3}\pi \)[/tex] cubic units and the radii we calculated:
[tex]\[
\left(\frac{r_{\text{small}}}{r_{\text{large}}}\right)^3 = \left(\frac{3}{5}\right)^3 = \frac{27}{125}
\][/tex]
Using the volume of the larger sphere:
[tex]\[
\text{Volume of Smaller Sphere} = \frac{27}{125} \times \frac{500}{3}\pi = \frac{27 \times 500}{125 \times 3}\pi = \frac{13500}{375}\pi = 36\pi \text{ cubic units}
\][/tex]
So, the volume of the smaller sphere is [tex]\( 113.09733552923254 \pi \)[/tex] cubic units.
In summary, we have:
- Scale factor = [tex]\( 1.6667 \)[/tex]
- Radius of smaller sphere = [tex]\( 3 \)[/tex] units
- Radius of larger sphere = [tex]\( 5 \)[/tex] units
- Volume of smaller sphere = [tex]\( 113.09733552923254 \)[/tex] cubic units
