Problema Solution
An offshore oil well is leaking oil and creating a circular oil slick. If the radius of the slick is growing at a rate of 2 miles/hour, find the rate at which the area is increasing when the radius is 5 miles.
Answer provided by our tutors
We assume that the leaking of the oil is circular.
We know that
v = 2 miles/hour is the rate of the radius growing
Also for v we have v = dr/dt ( d is derivative)
r = 5 miles - the radius of the circle
A = r^2 * Pi - the area of the circle
The rate at which the circular area is increasing will be derivative of the area with respect to time:
dA/dt = d( r^2 * Pi)/ dt
dA/dt = 2*Pi*r dr/dt
We put r = 5 and dr/dt = 2 and get
dA/dt = 2*Pi*5*2
dA/dt = 20 *Pi miles^2/hour
dA/dt = 62.8 miles^2/hour approximately
The rate at which the circular area is increasing is 62.8 miles^2/hour approximately.