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 by V(r) = 0.08πr², where r is the radius of the slick in feet. In turn, the radius is increasing over time according to the function r(t) = 0.5t, where t is measured in minutes. Find (V(r(t)), and give a practical interpretation of what this function tells you. After how many minutes will the volume of the slick be 705 cubic feet? Round your answer to the nearest minute.​

Answer 1

Solution :

We can start by substituting the expression for the radius given by r(t) = 0.5t into the expression for the volume of the oil slick V(r) = 0.08πr²:

V(r(t)) = 0.08π(0.5t)² = 0.02πt²

This tells us the volume of the oil slick as a function of time t. In other words, it gives us the amount of oil that is being spilled into the water as time passes.

To find out after how many minutes the volume of the slick will be 705 cubic feet, we can set V(r(t)) equal to 705 and solve for t:

0.02πt² = 705

Dividing both sides by 0.02π, we get:

t² = 705 / 0.02π

t² ≈ 56080.5

Taking the square root of both sides, we get:

t ≈ 237.0

Rounding this to the nearest minute, we get that after about 237 minutes (or 3 hours and 57 minutes), the volume of the slick will be 705 cubic feet.

The practical interpretation of the function V(r(t)) = 0.02πt² is that it gives the amount of oil spilled into the water as a function of time, taking into account the expansion of the oil slick as it spreads. This information is important for estimating the environmental impact of the oil spill and for determining the resources and strategies needed for cleanup.