Question

The radius of a spherical balloon is increasing at the rate of 0.9 cm/minute. How fast is the volume changing when the radius is 7.1 cm?

The volume is changing at a rate of ________ cm^3/minute
(Type an integer or a decimal Round to one decimal place as needed)

Answer 1

568.54 cm^3/minute when the radius is 7.1 cm.

Explanation:

To find how fast the volume is changing, we can use the relationship between the radius and the volume of a sphere. The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.

We are given that the radius is increasing at a rate of 0.9 cm/minute. We need to find the rate of change of the volume when the radius is 7.1 cm.

Let's differentiate the volume formula with respect to time:

dV/dt = (4/3)π(3r^2)(dr/dt)

Now we can substitute the given values:

r = 7.1 cm

dr/dt = 0.9 cm/minute

dV/dt = (4/3)π(3(7.1)^2)(0.9)

dV/dt = (4/3)π(3(50.41))(0.9)

dV/dt = (4/3)π(151.23)(0.9)

dV/dt = (4/3)(135.75)π

dV/dt = 181π

Calculating the numerical value:

dV/dt ≈ 568.54 cm^3/minute

Therefore, the volume is changing at a rate of approximately 568.54 cm^3/minute when the radius is 7.1 cm.