The radius of the large sphere is three times the radius of the small sphere.

How many times the volume of the large sphere is the volume of the small sphere?

A. [tex]$\frac{1}{27}$[/tex]
B. [tex]$\frac{1}{18}$[/tex]
C. [tex]$\frac{1}{9}$[/tex]
D. [tex]$\frac{1}{3}$[/tex]



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.