Question

Rates of change help. Step by step explanation needed. Will reward medal and fan. *question attached below*

Answer 1

you're given :
\(\large \frac{dA}{dt} = -2\)

Answer 2

\(A = \pi r^2\)

Answer 3

differentiate both sides with.respect.to t,
wat do u get ?

Answer 4

@ganeshie8 That's the step that I'm actually confused about

Answer 5

dA/dt = 2 pi r?

Answer 6

thats right if we're differentiating with.respect.to r,
however we're differentiating with.respect.to t

so by chain rule, u need to differentiate \(r\) again

Answer 7

\(\large A = \pi r^2\)


\(\large \frac{dA}{dt} = 2\pi r \frac{dr}{dt}\)

Answer 8

see if that looks okay,
next plugin the known values and solve \(\frac{dr}{dt}\)

Answer 9

I got -1/3?

Answer 10

lets see :)

we want to find dr/dt when the area is 9 pi, right ?

Answer 11

Yes

Answer 12

lets find out what the radius is, when area is 9 pi

\(A = \pi r^2\)

\(9 \pi = \pi r^2\)

solve \(r\)

Answer 13

r is 3

Answer 14

yes, so we wanto find dr/dt, when r = 3 :

\(\large \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \)

Answer 15

plugin r = 3, dA/dt = -2 above

Answer 16

\(\large \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \)

\(\large -2 = 2\pi *3* \frac{dr}{dt} \)

Answer 17

solve \(\frac{dr}{dt}\)

Answer 18

-1/3pi ?

Answer 19

\(\large \color{red}{\checkmark}\)

Answer 20

Thank you very much for your help and patience!
One question, though..how did we get dA/dt to be -2 when the rate was given as 2? @ganeshie8

Answer 21

u wlc :)

the area was given as decreasing, so... dA/dt = -2
if it was given as increasing, then it wud have been dA/dt = +2b

Answer 22

OOOOH..That makes sense..thanks :D

Answer 23

np... :)