Question

An accident at an oil drilling platform is causing a circular-shaped oil slick to form. The volume of the oil slick is roughly given V(r) = 0.07πr^2, where r is the radius of the slick in feet. In turn, the radius is increasing over time according to the function r(t) = 0.4t, where t is measured in minutes.

1) find (V of r)(t) and simply it

Answer 1

`(V(r))(t)=0.0112\pi t^(2)`

Explanation:

Given : An accident at an oil drilling platform is causing a circular-shaped oil slick to form. The volume of the oil slick is roughly given
`V(r) = 0.07\pi r^2`, where r is the radius of the slick in feet. In turn, the radius is increasing over time according to the function
`r(t)=0.4t` where t is measured in minutes.

To find : (V of r)(t) and simply it ?

Solution :

Let
`V(r) = 0.07\pi r^2` ....(1)

and
`r(t)=0.4t` ....(2)

For (V of r)(t)=V(r(t)) substitute equation (2) in (1),

i.e.
`V(r(t))=V(0.4t)`

`V(r(t))=0.07\pi (0.4t)^2`

`(V(r))(t)=0.07\pi (0.16)t^(2)`

`(V(r))(t)=0.0112\pi t^(2)`

Answer 2

V(t) = 0.0112πt^2

Explanation:

Substitute the expression for r and combine factors.

V(r(t)) = 0.07π(0.4t)^2

V(t) = 0.0112πt^2