To solve the problem of how many times the volume of the large sphere is the volume of the small sphere, we follow these steps:
1. Understand the relationship between radii:
The large sphere's radius is three times that of the small sphere. Let's denote the radius of the small sphere as [tex]\( r \)[/tex]. Therefore, the radius of the large sphere is [tex]\( 3r \)[/tex].
2. Formula for the volume of a sphere:
The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
3. Calculate the volume of the small sphere:
Let's assume the radius [tex]\( r \)[/tex] of the small sphere is 1 unit for simplicity. The volume of the small sphere is:
[tex]\[
V_{\text{small}} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi
\][/tex]
4. Calculate the volume of the large sphere:
The radius of the large sphere is [tex]\( 3r \)[/tex] (which is 3 in our case). The volume of the large sphere is:
[tex]\[
V_{\text{large}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi
\][/tex]
5. Find the ratio of the volumes:
To determine how many times the volume of the large sphere is compared to the volume of the small sphere, we take the ratio of the volume of the large sphere to the volume of the small sphere:
[tex]\[
\text{Volume Ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{36 \pi}{\frac{4}{3} \pi} = \frac{36 \pi \cdot 3}{4 \pi} = \frac{108 \pi}{4 \pi} = 27
\][/tex]
Therefore, the volume of the large sphere is 27 times the volume of the small sphere. The correct answer to the question is not listed among the given options, but our calculation indicates the volume ratio is:
[tex]\[
27
\][/tex]
