To determine the relationship between the volumes of two similar spheres, we can use the fact that the volume of a sphere is proportional to the cube of its radius.
The formula for the volume [tex]\( V \)[/tex] of a sphere with radius [tex]\( r \)[/tex] is given by:
[tex]\[ V = \frac{4}{3}\pi r^3 \][/tex]
Let's denote the radius of sphere B as [tex]\( r \)[/tex]. Therefore, its volume [tex]\( V_B \)[/tex] is:
[tex]\[ V_B = \frac{4}{3}\pi r^3 \][/tex]
According to the problem, the radius of sphere A is 3 times the radius of sphere B. Thus, the radius of sphere A is [tex]\( 3r \)[/tex]. The volume [tex]\( V_A \)[/tex] of sphere A is then:
[tex]\[ V_A = \frac{4}{3}\pi (3r)^3 \][/tex]
Now, let's calculate [tex]\( (3r)^3 \)[/tex]:
[tex]\[ (3r)^3 = 3^3 \cdot r^3 = 27r^3 \][/tex]
Therefore, the volume [tex]\( V_A \)[/tex] of sphere A is:
[tex]\[ V_A = \frac{4}{3}\pi \cdot 27r^3 = 27 \left( \frac{4}{3}\pi r^3 \right) = 27 V_B \][/tex]
This shows that the volume of sphere A is 27 times the volume of sphere B.
So, the correct answer is:
The volume of sphere A is 27 times the volume of sphere B.
