The volume of air inside a rubber ball with radius [tex]$r$[/tex] can be found using the function [tex]$V(n)=\frac{9}{3} \pi r$[/tex]. What does [tex]$V\left(\frac{9}{7}\right)$[/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 a volume function V ( r ) = 3 π r for a rubber ball with radius r .
We substitute r = 7 9 into the function to find V ( 7 9 ) .
The result V ( 7 9 ) = 3 π ( 7 9 ) = 7 27 π represents the volume of the rubber ball when the radius is 7 9 feet.
Therefore, V ( 7 9 ) represents the volume of the rubber ball when the radius equals 7 9 feet, which is 7 27 π cubic feet.
Explanation
Analyze the problem and the given function. The problem states that the volume of air inside a rubber ball with radius r is given by the function $V(r) =
3 9 π r = 3 π r
We are asked to determine what V ( 7 9 ) represents.
Substitute the given value into the function. To find out what V ( 7 9 ) represents, we substitute r = 7 9 into the volume function:
V ( 7 9 ) = 3 π ( 7 9 )
V ( 7 9 ) = 7 27 π
The result, V ( 7 9 ) = 7 27 π , represents the volume of the rubber ball when the radius is 7 9 feet.
Conclusion. Therefore, V ( 7 9 ) represents the volume of the rubber ball when the radius equals 7 9 feet.
Examples
Imagine you're inflating a spherical balloon. The function V ( r ) = 3 π r tells you how much air (volume) you need based on how far you've inflated it (the radius). If you inflate the balloon to a radius of 7 9 feet, then V ( 7 9 ) calculates the exact volume of air inside the balloon at that specific radius. This is useful in manufacturing to ensure balloons have the correct size and volume for their intended use.
The expression V ( 7 9 ) represents the volume of the rubber ball when the radius is 7 9 feet, which can be calculated to be 7 27 π cubic feet. The correct choice from the multiple-choice options is B . This indicates the relationship between the radius and the ball's volume.
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