A stone is dropped into a quiet lake and waves move in the form of circles at a speed of 4 cm/sec. At the instant, when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?

Dear Student

We know that the area of a circle with radius “r” is given by A = πr^2.

Hence, the rate of change of area A with respect to the time t is given by:

dA/dt = (d/dt) πr^2

By using the chain rule, we get:

(d/dr)(πr^2). (dr/dt) = 2πr.(dr/dt)

It is given that, dr/dt = 4 cm/sec

Therefore, when r = 10 cm,

dA/dt = 2π. (10). (4)

dA.dt = 80 π

Hence, when r = 10 cm,
the area is increasing at a rate of 80π cm^2/sec.

Thanks