Question

the radius (r) of a circle is increasing at a rate of 4 cm per minute. what is the rate of change of the area when r=8 cm

Answer 1

So you need to come up with a formula that involves all the info given. The question tells you in the problem itself, we need a circle formula. So that would be
\[A = \pi r^{2}\]Once you have the proper formula, you take the derivative of it and label it like this:
\[\frac{ dA }{ dt }=2\pi r*\frac{ dr }{ dt }\]
Every variable you take the derivative of comes with its own dr/dt, da/dt, ds/dt, etc. So I had the unknown variable A and the unknown variable r, therefore I have an appropriate Da/dt and dr/dt for each.

From here, I just plug in information. Anything to do with change involves the d/dt terms. So when it says the radius is increasing at a rate of 4cm per minute, this means that dr/dt is 4. So we're given dr/dt and we're given r = 8, so now we solve for the change in the area, or simply da/dt

\[\frac{dA}{dt} = 2 \pi (8)(4) \implies \frac{dA}{dt} = 64 \pi\]

Answer 2

wow thank you for the nice explanation

Answer 3

yeah, np :3