The area of a circle is increasing at a rate of 12pi ft^2/sec. At what rate is the radius of 6ft. increasing?
The engineer's way.
As the radius grows by dr, the area grows by the circumference times dr. That is
dA = C*dr = 2πr dr
so, when dA=12 and r=6, dr=1
Or, as they want you to use calculus,
A = πr^2
dA/dt = 2πr dr/dt
12π = 2π*6 * dr/dt
dr/dt = 1 ft/s