To determine how many times larger the volume of the large sphere is in comparison to the volume of the small sphere, let's go through the problem step by step.
1. Volume of a Sphere Formula:
The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere.
2. Volume of the Small Sphere:
Let [tex]\( r \)[/tex] be the radius of the small sphere. The volume of this small sphere ([tex]\( V_{\text{small}} \)[/tex]) is:
[tex]\[
V_{\text{small}} = \frac{4}{3} \pi r^3
\][/tex]
3. Volume of the Large Sphere:
The radius of the large sphere is three times the radius of the small sphere. Therefore, the radius of the large sphere is [tex]\( 3r \)[/tex]. The volume of the large sphere ([tex]\( V_{\text{large}} \)[/tex]) is:
[tex]\[
V_{\text{large}} = \frac{4}{3} \pi (3r)^3
\][/tex]
Simplifying [tex]\( (3r)^3 \)[/tex]:
[tex]\[
(3r)^3 = 27r^3
\][/tex]
So the volume of the large sphere becomes:
[tex]\[
V_{\text{large}} = \frac{4}{3} \pi (27r^3)
\][/tex]
[tex]\[
V_{\text{large}} = 27 \left( \frac{4}{3} \pi r^3 \right)
\][/tex]
[tex]\[
V_{\text{large}} = 27 V_{\text{small}}
\][/tex]
4. Ratio of Volumes:
To find how many times the volume of the large sphere is larger than the volume of the small sphere, we compute the ratio [tex]\( \frac{V_{\text{large}}}{V_{\text{small}}} \)[/tex]:
[tex]\[
\frac{V_{\text{large}}}{V_{\text{small}}} = \frac{27 V_{\text{small}}}{V_{\text{small}}} = 27
\][/tex]
Thus, the volume of the large sphere is 27 times larger than the volume of the small sphere.
Answer:
[tex]\[
\boxed{27}
\][/tex]
