Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 2m/s, how fast is the area of the spill increasing when the radius is 15m?
To find how fast the area of the oil spill is increasing, we can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.
We are given that the radius is increasing at a constant rate of 2 m/s. Let's denote the rate at which the radius is increasing as dr/dt.
We want to find how fast the area is increasing when the radius is 15 m. Let's denote the rate at which the area is increasing as dA/dt.
Now, let's differentiate the equation A = πr^2 with respect to time (t) using the chain rule:
dA/dt = d/dt(πr^2)
= 2πr(dr/dt)
We know that dr/dt = 2 m/s, and we want to find dA/dt when r = 15 m. Substituting the given values:
dA/dt = 2π(15)(2)
= 60π m^2/s
Therefore, when the radius is 15 m, the area of the spill is increasing at a rate of 60π m^2/s.
d/dt(pi*r^2) = 2*pi*r*dr/dt
Do the numbers.