The volume V (in cubic meters) of the hot-air balloon described in problem 65 is given by V(r) = (4)/(3) \pi r^(3). If the radius r is the same function as t as in problem 65, f…

Question

The volume V (in cubic meters) of the hot-air balloon described in problem 65 is given by
`V(r) = (4)/(3) \pi r^(3)`. If the radius r is the same function as t as in problem 65, find the volume V as a function of the time t. Use
`r(t) = (2)/(3) t^(2)` to solve.

Answer 1

`V(t)=(32)/(81)\pi t^6`

Explanation:

You are given two functions

`V(r)=(4)/(3)\pi r^3\\ \\r(t)=(2)/(3)t^2`

You have to find the volume V as a function of the time t. Substitute the expression of r into the function V(r) to get V(t):

`V(t)\\ \\=(4)/(3)\pi\cdot \left((2)/(3)t^2\right)^3\\ \\=(4)/(3)\pi \cdot (2^3)/(3^3)(t^2)^3\\ \\=(4)/(3)\pi\cdot (8)/(27)t^6\\ \\=(32)/(81)\pi t^6`