The function [tex]$V(r)=\frac{4}{3} \pi r^3[tex]$[/tex] can be used to find the volume of air inside a basketball given its radius. What does [tex]$[/tex]V(r)$[/tex] represent?

A. The radius of the basketball when the volume is [tex]$V$[/tex]
B. The volume of the basketball when the radius is [tex]$r$[/tex]
C. The volume of the basketball when the radius is [tex]$V$[/tex]
D. The radius of the basketball when the volume is [tex]$r$[/tex]



Answer :

The function \( V(r) = \frac{4}{3} \pi r^3 \) is used to calculate the volume of a sphere, in this case, a basketball, based on its radius \( r \).

Let's break down the function step by step:

1. Formula Explanation:
- The formula \( V(r) = \frac{4}{3} \pi r^3 \) calculates the volume of a sphere.
- Here, \( r \) is the radius of the sphere (basketball), and \( \pi \) is a mathematical constant approximately equal to 3.14159.

2. Term Identification:
- \( V(r) \) represents the volume of the sphere as a function of its radius \( r \).
- The radius \( r \) is the distance from the center of the basketball to any point on its surface.

3. Variable Roles:
- In this function, \( r \) is the independent variable (input), and \( V(r) \) is the dependent variable (output).
- By substituting a specific value of \( r \) into the formula, we get the corresponding volume \( V(r) \).

4. Interpretation of \( V(r) \):
- \( V(r) \) is essentially saying "the volume of the basketball when the radius is \( r \)."

5. Answer Selection:
- Given the options, the correct interpretation of what \( V(r) \) represents is:
- The volume of the basketball when the radius is \( r \).

Hence, [tex]\( V(r) = \frac{4}{3} \pi r^3 \)[/tex] represents the volume of the basketball when the radius is [tex]\( r \)[/tex].