Question

A spherical balloon is being filled with air. The radius of the balloon is given by the function R(t) = 3t, where t represents time in seconds. The volume of the balloon is given by the function V(r) = 4/3 πr³, where r represents the radius in inches. Which function can be used to determine the volume of the balloon as a function of time?

A) πR(V(t))
B) R(V(t))
C) πV(R(t))
D) V(R(t))

Answer 1

The correct function to determine the volume of the balloon as a function of time is V(R(t)), which translates into V(t) = 36πt³, representing the volume of the balloon as time changes.

Step-by-step explanation:

The student is asking about how to express the volume of a balloon as a function of time, using the given formulas for the radius and volume of a sphere.

We have a function for the radius of the balloon R(t) = 3t, where t is time, and a function for the volume of a sphere V(r) = 4/3 πr³, where r is the radius.

To find the function that represents the volume of the balloon as a function of time, we need to substitute the expression for R(t) into the volume function V(r). This gives us V(R(t)) = 4/3 π(3t)³. Therefore, the correct option is D) V(R(t)).

Upon evaluating, we get V(t) = 4/3 π(27t³) = 36πt³, which is the volume of the balloon as a function of time.