Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mi^2/hr. How rapidly is radius of the spill increasing when the area is 9 mi^2?

Question

Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 mi^2/hr. How rapidly is radius of the spill increasing when the area is 9 mi^2?

The radius is increasing at _____ mi/hr.

Answer 1

When the radius area is 9 square miles, the radius can be determined by A = pi*r^2. So 9 = pi*r^2 implies that r = 3/sqrt(pi). Keep that in mind for a moment.

The area of a circle is A = Pi*r^2. Since the rate of change in area is given in the problem dA/dt = 6 mi^2/h.

Differentiate both sides with respect to t:

dA/dt = Pi*2*r*dr/dt (by the chain rule).

From above, r = 3/sqrt(pi), dA/dt is 6, solve for dr/dt (which is what the problem is asking). Good luck.