Question

A stone is dropped into a quiet lake and waves in the form of circles are generated. Radius of the circular wave increases at the rate of 3 cm/sec. How fast the area enclosed is increasing when the radius is 8 cm?

Answer 1

Given:

Rate of increase of radius

`(dr)/dt` = 3 cm/sec

We need to find:

`(dA)/dt` when r = 8 cm

Step 1: Formula for area of circle

A = πr²

Step 2: Differentiate with respect to time t

`(dA)/dt = π * 2r (dr)/dt`

`(dA)/dt = 2πr (dr)/dt`

Step 3: Substitute values

`(dA)/dt = 2π(8)(3)`

`(dA)/dt` = 48π cm²/sec