Question

The radius of an air bubble is increasing at the rate of cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Answer 1

The radius of the air bubble is increasing at the rate of 1 2   cm/s . The radius of the air bubble is 1 cm.

The shape of the air bubble is sphere.

Let the radius of air bubble be r and the volume of sphere be V.

The volume of the air bubble is given by,

V= 4 3 π r 3 (1)

Differentiate equation (1) to obtain the rate of change of V with respect to time t.

dV dt = 4 3 π d dr r 3 ⋅ dr dt = 4 3 π( 3 r 2 )⋅ dr dt =4π r 2 ⋅ dr dt (2)

According to the given condition,

dr dt = 1 2   cm/s

Substitute the value of rand dr dt in equation (2).

dV dt =4π ( 1 ) 2 ⋅ 1 2 =2π  cm 2 /s

Thus, the volume of bubble is increasing at the rate of 2π  cm 2 /s .