The volume of air inside a rubber ball with radius [tex]$r$[/tex] can be found using the function [tex]$v(r) = \frac{4}{3} \pi r^3$[/tex]. What does [tex]$v(\frac{5}{7})$[/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 formula for the volume of a sphere: v ( r ) = 3 4 π r 3 .
We are asked to interpret v ( 7 5 ) .
Substituting r = 7 5 into the formula gives the volume when the radius is 7 5 feet.
Therefore, v ( 7 5 ) represents the volume of the rubber ball when its radius is 7 5 feet, which is approximately 1.527 cubic feet. the volume of the rubber ball when the radius equals 7 5 feet
Explanation
Analyze the problem 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.
Substitute the value To find out what v ( 7 5 ) represents, we substitute r = 7 5 into the function v ( r ) . This gives us v ( 7 5 ) = 3 4 π ( 7 5 ) 3 .
Determine the meaning The expression v ( 7 5 ) represents the volume of the rubber ball when the radius is 7 5 feet. The result of the calculation is approximately 1.527 cubic feet.
Examples
Understanding the volume of spheres is crucial in various fields. For example, when designing spherical containers for liquids or gases, engineers need to accurately calculate the volume to ensure the container meets the required capacity. Similarly, in sports, knowing the volume of a ball helps in determining its mass and aerodynamic properties, which are essential for optimal performance. In medical applications, calculating the volume of spherical tumors aids in monitoring their growth and planning treatment strategies. The formula v ( r ) = 3 4 π r 3 is a fundamental tool in these calculations.
The expression v ( 7 5 ) represents the volume of the rubber ball when its radius is 7 5 feet. Therefore, the correct answer is option B. The volume of the rubber ball when the radius equals 7 5 feet.
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