A spherical balloon is inflated with gas at the rate of cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is centimeters?

Question

A spherical balloon is inflated with gas at the rate of  cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is  centimeters?

Quizplus · Accepted Answer

We know that the volume of a sphere is given by the formula:

V = (4/3)πr^3

Differentiating both sides with respect to time, we get:

dV/dt = 4πr^2(dr/dt)

We are given that dV/dt = 11eaa595_6614_5515_a696_0d89a924bebd_TB4584_11_TB4584_11 cubic centimeters per minute, and we need to find the value of (dr/dt) when r = 11eaa595_6614_5516_a696_6dcc943694e7_TB4584_11_TB4584_11 centimeters.

Substituting the given values, we get:

11eaa595_6614_5515_a696_0d89a924bebd_TB4584_11_TB4584_11 = 4π(11eaa595_6614_5516_a696_6dcc943694e7_TB4584_11_TB4584_11)^2(dr/dt)

Simplifying, we get:

(dr/dt) = 11eaa595_6614_ca49_a696_8d7e32ef4eec_TB4584_11_TB4584_11

Therefore, the answer is (C).

A Spherical Balloon Is Inflated with Gas at the Rate