The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Given: radius of an air bubble is increasing at the rate of 0.5 cm/sec

To find the rate at which the volume of the bubble increasing when the radius is 1 cm

Let the radius of the given air bubble be r cm and let V be the volume of the air bubble at any instant time

Then according to the given criteria,

Rate of increase in the radius of the air bubble is,

We know volume of the air bubble is .

Applying derivative with respect to time on both sides we get,

[from equation(i)]

So when the radius is 1cm, the above equation becomes,

Hence the rate at which the volume of the air bubble is increasing when the radius is 1 cm will be 2 cm³/sec.