Answer :
To find how many times the volume of the large sphere is compared to the volume of the small sphere, we can follow these steps:
1. Define the radius of the small sphere:
Let the radius of the small sphere be [tex]\( r \)[/tex].
2. Define the radius of the large sphere:
According to the problem, the radius of the large sphere is three times that of the small sphere. So, the radius of the large sphere is [tex]\( 3r \)[/tex].
3. Calculate the volume of the small sphere:
The formula for the volume [tex]\( V \)[/tex] of a sphere is given by
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
So, the volume of the small sphere with radius [tex]\( r \)[/tex] is:
[tex]\[ V_{\text{small}} = \frac{4}{3} \pi r^3 \][/tex]
4. Calculate the volume of the large sphere:
Using the same formula, the volume of the large sphere with radius [tex]\( 3r \)[/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 is:
[tex]\[ V_{\text{large}} = \frac{4}{3} \pi \times 27r^3 = 27 \left( \frac{4}{3} \pi r^3 \right) \][/tex]
Notice that [tex]\( 27 \times \frac{4}{3} \pi r^3 \)[/tex] simplifies to:
[tex]\[ V_{\text{large}} = 27 \times V_{\text{small}} \][/tex]
5. Compare the volumes:
From the calculation above, we can see that the volume of the large sphere is 27 times the volume of the small sphere.
Therefore, the correct answer is:
[tex]\[ \boxed{27} \][/tex]
However, since the options provided are [tex]\(\frac{1}{27}\)[/tex], [tex]\(\frac{1}{18}\)[/tex], [tex]\(\frac{1}{9}\)[/tex], and [tex]\(\frac{1}{3}\)[/tex], and they seem to be inverse values rather than direct multiples, none of these options match the correct factor of 27. Thus, the provided options might need re-evaluation as they do not include the correct choice.
1. Define the radius of the small sphere:
Let the radius of the small sphere be [tex]\( r \)[/tex].
2. Define the radius of the large sphere:
According to the problem, the radius of the large sphere is three times that of the small sphere. So, the radius of the large sphere is [tex]\( 3r \)[/tex].
3. Calculate the volume of the small sphere:
The formula for the volume [tex]\( V \)[/tex] of a sphere is given by
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
So, the volume of the small sphere with radius [tex]\( r \)[/tex] is:
[tex]\[ V_{\text{small}} = \frac{4}{3} \pi r^3 \][/tex]
4. Calculate the volume of the large sphere:
Using the same formula, the volume of the large sphere with radius [tex]\( 3r \)[/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 is:
[tex]\[ V_{\text{large}} = \frac{4}{3} \pi \times 27r^3 = 27 \left( \frac{4}{3} \pi r^3 \right) \][/tex]
Notice that [tex]\( 27 \times \frac{4}{3} \pi r^3 \)[/tex] simplifies to:
[tex]\[ V_{\text{large}} = 27 \times V_{\text{small}} \][/tex]
5. Compare the volumes:
From the calculation above, we can see that the volume of the large sphere is 27 times the volume of the small sphere.
Therefore, the correct answer is:
[tex]\[ \boxed{27} \][/tex]
However, since the options provided are [tex]\(\frac{1}{27}\)[/tex], [tex]\(\frac{1}{18}\)[/tex], [tex]\(\frac{1}{9}\)[/tex], and [tex]\(\frac{1}{3}\)[/tex], and they seem to be inverse values rather than direct multiples, none of these options match the correct factor of 27. Thus, the provided options might need re-evaluation as they do not include the correct choice.
