Question

Rates of Change/Differentiation question help. Will reward medal. *question attached below*

Answer 1

Great question. First, find the radius of the circle with area of 9pi units^2

Answer 2

How do I do this? @iPwnBunnies

Answer 3

Do you know the formula for the area of a circle?

Answer 4

You should know if you're doing Calculus lol.

Answer 5

Area = pi*r^2

Answer 6

Yes, I know this..but I was having trouble deriving the radius from the area given :S

Answer 7

I get it. We're taking it one step at the time. What's the radius of the circle?

Answer 8

We need the radius of the circle in the end.

Answer 9

A = pi*r^2; We will need to derive this equation with respect to t, time. That's what related rates are.
Sooooo, can you find the derivative of the area formula?

Answer 10

i dont know if he deleted some of his comments. but i dont think discouraging him will do him any good.

Answer 11

Well, he's not even here anymore. He hasn't been here since I told him we need the radius. His downfall.

Answer 12

I didn't even delete any comments.

Answer 13

My internet have been giving me problems.

Answer 14

And it's a her.

Answer 15

The first step is write down the equation
\[ Area = \pi r^2 = 9 \pi \ cm^2 \]
next, solve for r
can you do that ?

Answer 16

Sorry.

Answer 17

I tried to get r but I think the 9 pi in the equation of the area is making it difficult

Answer 18

you have
\[ \pi r^2 = 9 \pi \]
divide both sides by pi
then take the square root of both sides
what do you get ?

Answer 19

I ended up with 3

Answer 20

\[\Pi(3)^{2}=9\Pi\]

Answer 21

you are right

Answer 22

If you are doing calculus, you should know your algebra inside out and backwards. Otherwise you will get stuck.

Next, take the derivative with respect to time
\[ \frac{d}{dt}( A = \pi r^2) \\ \frac{d}{dt}A= \pi \frac{d}{dt}r^2 \]

Answer 23

I am quite confused now :S

Answer 24

do you know how to take the derivative of
y= x
with respect to x ?
\[ \frac{d}{dx} ( y=x) \\ \frac{d}{dx}y = \frac{d}{dx} x \]

Answer 25

I know the answer for the derivative of y is equal to 1?

Answer 26

yes, and we write it
\[ \frac{d}{dx} ( y=x) \\ \frac{d}{dx}y = \frac{d}{dx} x \\ \frac{dy}{dx}= \frac{dx}{dx} \\ \frac{dy}{dx}= 1 \]

in your problem where both A and r change with time (i.e. are functions of time) we follow the same procedure
\[ \frac{d}{dt}A= \pi \frac{d}{dt}r^2\\ \frac{dA}{dt}= \pi \ 2r\frac{dr}{dt} \]
now fill in the variables you know: r= 3, dA/dt= -20
and solve for dr/dt, which is how fast the radius is decreasing