The volume of a sphere is increasing at a constant rate of 5741 cubic inches per minute. At the instant when the volume of the sphere is 1770 cubic inches,

Question

What is the rate of change of the surface area of the sphere? The volume of a sphere can be found with the equation V = 4/3 π r^3 and the surface area can be found with S = 4 π r^2. Round your answer to three decimal places (if necessary).

Answer 1

The volume of the sphere formula is given to be V = 4/3 pi * r^3

It gives you that the rate the volume is increasing is 5471 cubic inches per minute. This rate is your derivative dV/dt.

Taking the derivative of the volume formula

DV/dt = 4 pi r*2 dr/dt

Plugging in 5471 for dV/dt gets

5471 = 4 pi r*2 dr/dt

This equation has two unknowns, r and dr/dt

We can solve for r...

You are told the volume of the sphere is 1770 cubic inches.

Setting this equal to V

1770 = 4/3 pi r^3

You can now solve for r.

One you solved for r, plug it back in to 5471 = 4 pi r*2 dr/dt

And you can now solve for dr/dt, the rate at which the radius is changing with respect to time.

Now, to find the rate the surface area is changing, you must take the derivative of the surface area function.

S = 4 pi r^2

dS/dt = 8 pi r dr/dt

You have solved to find both r and dr/dt. You can plug these values into the formula above to solve for dS/dt, the rate the surface area is changing with respect to time. This will be your answer.

I hope this helps get you started.