The area of a circle is increasing at a rate of 4ft /min.How fast is the radius of this circle increasing at the instant the area is 16π ft?

Answer 1

Area (A) of a circle is = π R²  where R = radius of the circle.

=>  A  =  π R²

When Area  =  16π,  R = 4 ft  ......................... (1)

Differentiate both the sides with respect to Time (t),

dA/dt = 2 π R dR/dt

dA = Rate of change in area = 4 ft² /min  and Radius =  4 ft ..... (from 1 above )

=>  Rate of change in Area  =  ( 2 π R ) *  Rate in change of Radius.

=>  4  =  2 π 4 ( dR/dt)

=> dR/dt  =  1/(2π)  ft/min ........ Answer

Answer 2

dA/dt = 4

A = pi * r^2

dA/dt = 2 * pi * r * dr/dt

A = 16 * pi

16 * pi = pi * r^2

16 = r^2

4 = r

dA/dt = 2 * pi * r * dr/dt

4 = 2 * pi * 4 * dr/dt

1/(2pi) = dr/dt