Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mi2/hr. How rapidly is radius of the spill increasing when the area is 10mi2?

Question

Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mi²/hr. How rapidly is radius of the spill increasing when the area is 10mi²?

Answer 1

1) Write down what you know:

Area increases at a constant rate of 6 mi^2/hr: dA/dt = 6

dr/dt = ?

A = π r ²

2) Take the derivative with respect to time:

dA/dt = 2 π r * (dr/dt)

3) Substitute values and solve for dr/dt:

When A = 10 -> 10 = π r² -> r = sqrt(10/pi)

6 = 2 π sqrt(10/pi) * (dr/dt)

dr/dt = 3/(π*sqrt(10/pi)) miles/hr .535 miles/hr