The volume of air inside a rubber ball with radius [tex]$r$[/tex] can be found using the function [tex]$V(n)=\frac{4}{3} \pi r^3$[/tex]. What does [tex]$V\left(\frac{5}{7}\right)$[/tex] represent?
A. the radius of the rubber ball when the volume equals [tex]$\frac{5}{7}$[/tex] cubic feet
B. the volume of the rubber ball when the radius equals [tex]$\frac{5}{7}$[/tex] feet
C. that the volume of the rubber ball is 5 cubic feet when the radius is 7 feet
D. that the volume of the rubber ball is 7 cubic feet when the radius is 5 feet
Answer (2)
The problem provides the volume function V ( r ) = 3 4 π r 3 for a rubber ball with radius r .
We substitute r = 7 5 into the volume function to get V ( 7 5 ) = 3 4 π ( 7 5 ) 3 .
V ( 7 5 ) represents the volume of the rubber ball when its radius is 7 5 feet.
Therefore, the answer is: \boxed{\text{the volume of the rubber ball when the radius equals \frac{5}{7} feet}} .
Explanation
Problem Analysis The problem states that the volume of air inside a rubber ball with radius r is given by the function V ( r ) = 3 4 π r 3 . We are asked to determine what V ( 7 5 ) represents.
Substitution To understand what V ( 7 5 ) means, we need to substitute r = 7 5 into the volume function. This gives us V ( 7 5 ) = 3 4 π ( 7 5 ) 3 .
Interpretation The expression V ( 7 5 ) represents the volume of the rubber ball when its radius is 7 5 feet.
Conclusion Therefore, V ( 7 5 ) represents the volume of the rubber ball when the radius equals 7 5 feet.
Examples
Imagine you're designing inflatable beach balls. You need to know how much air each ball will hold. If you decide the radius of the ball should be 7 5 feet, you can use the formula V ( r ) = 3 4 π r 3 to calculate the volume of air needed. This helps you determine the size of the container needed to store the air and the amount of material required to make the ball.
The expression V ( 7 5 ) represents the volume of a rubber ball when the radius is 7 5 feet. Therefore, the correct answer is option B. This means that when substituting the radius into the volume formula, we are calculating the volume for that particular radius value.
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