A circle's radius is increasing at 0.1 feet/minute. How rapidly is its area increasing when the radius is 3 feet?

Question

A circle's radius is increasing at  feet/minute. How rapidly is its area increasing when the radius is  feet?

I followed this website's steps on how to solve this problem using the numbers given to me.

https://socratic.org/questions/if-the-radius-of-a-sphere-is-increasing-at-a-rate-of-4-cm-per-second-how-fast-is

I got 11.3097 ft^2/min for my answer which is wrong :( please help

Answer 1

Good Question!

Let's first look at the equation for the area of a circle:

A=π r²

Now the question is asking for the rate of change of the Area, so lets apply d/dt to both sides of the equation so we can find dA/dt (the rate of change of the area with respect to time):

dA/dt = d/dt (π r²)

Using Chain Rule where d/dt (r²) = 2r*dr/dt

dA/dt = π 2 r (dr/dt)

So, now we need to know what r is and what dr/dt is.

Thankfully, we were given these!

We are looking for the rate of change of the Area when r=3 ft and we know dr/dt = 0.1 ft/min

Let's plug those in:

dA/dt = 2π (3 ft) (0.1 ft/min)

Thus,

dA/dt = 6ft π(0.1 ft/min) ≈ 1.885 sqr ft/min

Answer 2

A=πr²

^{dA/dt = 2πr dr/dt}

^{In this case dA/dt = 6π*.1 which is approximately 1.88 sq/min}