The function [tex]$V(r)=\frac{4}{3} \pi r^3$[/tex] can be used to find the volume of air inside a basketball given its radius. What does [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 (2)
The function V ( r ) calculates the volume of a sphere based on its radius r .
r is the radius of the basketball.
V ( r ) is the volume of the basketball when the radius is r .
Therefore, V ( r ) represents the volume of the basketball when the radius is r . the volume of the basketball when the radius is r
Explanation
Analyze the function V(r) The function V ( r ) = 3 4 π r 3 gives the volume of a sphere (in this case, a basketball) as a function of its radius r . In this function:
r is the input, representing the radius of the basketball.
V ( r ) is the output, representing the volume of the basketball corresponding to the radius r .
Therefore, V ( r ) represents the volume of the basketball when the radius is r .
Determine what V(r) represents The function V ( r ) takes the radius r as an input and calculates the volume of the basketball. So, V ( r ) gives you the volume of the basketball for a given radius r .
Final Answer Therefore, V ( r ) represents the volume of the basketball when the radius is r .
Examples
Imagine you're inflating a basketball. As you pump air in, the basketball's radius increases. The function V ( r ) = 3 4 π r 3 allows you to calculate the volume of air inside the basketball at any given radius. For example, if the radius is 10 cm, the volume would be V ( 10 ) = 3 4 π ( 10 ) 3 ≈ 4188.79 cubic centimeters. This concept is useful in various real-world applications, such as designing sports equipment or calculating the capacity of spherical containers.
The function V ( r ) = 3 4 π r 3 calculates the volume of a basketball based on its radius r . Thus, V ( r ) represents the volume of the basketball when the radius is r . The correct answer is Option B.
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