Question

the function ( v(r) = \frac{4}{3}pi r^3 ) can be used to find the volume of air inside a basketball given its radius. what does ( v(r) ) represent?

the radius of the basketball when the volume is ( v )

the volume of the basketball when the radius is ( r )

the volume of the basketball when the radius is ( v )

the radius of the basketball when the volume is ( r )

Explanation:

The function is \( V(r)=\frac{4}{3}\pi r^{3} \), where \( r \) is the radius of the basketball. In a function \( V(r) \), the input is \( r \) (radius) and the output is \( V \) (volume). So \( V(r) \) represents the volume of the basketball when the radius is \( r \). Let's analyze each option:

• Option 1: If we wanted the radius when volume is \( V \), we would solve \( V = \frac{4}{3}\pi r^{3} \) for \( r \), and the function would be in terms of \( V \), not \( r \) as the input here. So this is incorrect.
• Option 2: Since \( r \) is the radius and we plug \( r \) into the function to get \( V \) (volume), this matches the definition of \( V(r) \).
• Option 3: The variable for radius is \( r \), not \( V \), so this is a mix - up of variables. Incorrect.
• Option 4: The input is radius (\( r \)) and output is volume (\( V \)), not radius as output with volume as input in this function. Incorrect.

Answer:

B. the volume of the basketball when the radius is \( r \)