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Relationships Between Lengths, Areas, and Volumes: Mastery Test
Select the correct answer from the drop-down menu.
Sphere A is similar to sphere B.
If the radius of sphere A is 3 times the radius of sphere B, then the volume of sphere A is
times the volume of sphere B.
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Verified Answer :

GinnyAnswer

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 \(\( V \)\) of a sphere with radius \(\( r \)\) is given by:
\(\[ V = \frac{4}{3}\pi r^3 \]\)

Let's denote the radius of sphere B as \(\( r \)\). Therefore, its volume \(\( V_B \)\) is:
\(\[ V_B = \frac{4}{3}\pi r^3 \]\)

According to the problem, the radius of sphere A is 3 times the radius of sphere B. Thus, the radius of sphere A is \(\( 3r \)\). The volume \(\( V_A \)\) of sphere A is then:
\(\[ V_A = \frac{4}{3}\pi (3r)^3 \]\)

Now, let's calculate \(\( (3r)^3 \)\):
\(\[ (3r)^3 = 3^3 \cdot r^3 = 27r^3 \]\)

Therefore, the volume \(\( V_A \)\) of sphere A is:
\(\[ V_A = \frac{4}{3}\pi \cdot 27r^3 = 27 \left( \frac{4}{3}\pi r^3 \right) = 27 V_B \]\)

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.

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