Question

A circle with area A = pi r^2 is expanding with time. If dA/dt = 5cm^2/s, what is dr/dt when r=6?

Answer 1

Hint:\[\frac{dA}{dt} = \frac{d}{dt}A = \frac{d}{dt}(\pi r^2)\]

Answer 2

so it would be 2r?

Answer 3

no, 2pi r dr/dt. (chain rule)

Answer 4

so now what?

Answer 5

\[\frac{dA}{dt} = \pi \frac{d}{dt}r^2 = \pi \frac{dr}{dt}(\frac{d}{dr}r^2) = \pi \frac{dr}{dt}2r\]Do you understand up to this point?

Answer 6

\[A = \pi r^2\\\frac{ dA }{ dt } = \frac{ d }{ dt }(\pi r^2) = 2\pi r \frac{ dr }{ dt }\]

Answer 7

Just plug in the numbers: dA/dt = 5, r = 6, find dr/dt

Answer 8

how would I find dr and dt?

Answer 9

Do you know what differentiation is?

Answer 10

sort of

Answer 11

It seems to me like maybe you get how to actually differentiate but you don't understand what exactly the result you're getting means. Something like dx/dt describes the relationship between two variables; it's not just an actual fraction that you can/would want to split up (in this context at least). For this particular example, dr/dt is the rate of change of the radius with respect to time, so all you have to do is evaluate "dr/dt" when r=6 if dA/dt = 5.