To solve this problem, let's first recall the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
where [tex]\( V \)[/tex] is the volume and [tex]\( r \)[/tex] is the radius of the sphere.
### Step-by-Step Solution:
1. Define the Radii:
- Let the radius of the small sphere be [tex]\( r \)[/tex].
- Since the radius of the large sphere is three times that of the small sphere, the radius of the large sphere will be [tex]\( 3r \)[/tex].
2. Calculate the Volume of the Small Sphere:
- Using the formula for the volume of a sphere:
[tex]\[ V_{\text{small}} = \frac{4}{3} \pi r^3 \][/tex]
3. Calculate the Volume of the Large Sphere:
- Similarly, for the large sphere with radius [tex]\( 3r \)[/tex]:
[tex]\[ V_{\text{large}} = \frac{4}{3} \pi (3r)^3 \][/tex]
- Simplify the expression:
[tex]\[ V_{\text{large}} = \frac{4}{3} \pi (27r^3) \][/tex]
[tex]\[ V_{\text{large}} = 27 \cdot \frac{4}{3} \pi r^3 \][/tex]
[tex]\[ V_{\text{large}} = 27 \cdot V_{\text{small}} \][/tex]
Hence, the volume of the large sphere is 27 times the volume of the small sphere.
4. Determine the Ratio of the Volumes:
- The question asks for the ratio of the volume of the large sphere to the volume of the small sphere.
- As derived:
[tex]\[ \text{Volume ratio} = \frac{V_{\text{large}}}{V_{\text{small}}} = 27 \][/tex]
Therefore, the volume of the large sphere is 27 times the volume of the small sphere.
Given the multiple-choice options are fractions, this affirmation suggests we need the inverse of the ratio to see if the answer fits one of the provided choices. The reciprocal of 27 is [tex]\(\frac{1}{27}\)[/tex].
So, the correct choice is:
[tex]\(\frac{1}{27}\)[/tex]
This means:
[tex]\[\text{The correct answer is} \ \boxed{\frac{1}{27}}.\][/tex]
