# How to solve calculus involving rates of a circle ?

Suppose a forest fire speads in a circle with radius changing at a rate of 5 feet per minute. When the radius reaches 200 feet, at what rate is the area of the burning region increasing?

i kno the answer is 2000pi

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• Anonymous

A = pi R^2

dA/dt = 2pi R (dR/dt) = 2pi(200)(5) = 2000pi ft^2/min

• A H
Lv 6

With these related rates problems, you first need a formula that works all the time. In this case, your formula for area should qualify:

A = pi * r^2

Once you get that formula, you need to look at what's changing, and what it's changing with respect to (usually the latter is time). Then, you differentiate with respect to that variable - time. So, differentiating with respect to time, you get:

dA/dt = d/dt (pi * r^2)

Now, looking at the left, area obviously changes wrt time, so you should have a nonzero value on the right. The number pi doesn't change with respect to time (although our estimates do - google "Indiana" and "pi"!), so that's just a constant that can get pulled out.

dA/dt = pi * d/dt (r^2)

Now you have to ask if the radius changes with respect to time. Well, the problem SAYS it does! Thus, dr/dt is NOT zero. So, you need to differentiate r^2 with respect to t, using the chain rule. You get:

dA/dt = pi * 2r * dr/dt

You have all the info in the problem. Notice anything fun about that equation? Like the fact that 2pi * r is in it? Ask your teacher to draw a diagram (if you can't) to show why this is true!

you are given rate at which radius changes with respect to time is 5feet/min

dr/dt = 5

you need to find dA/dt ( A= area)

dA/dt = dr/dt * dA/dr

A = pi*r^2

dA/dr = 2*pi*r

dA/dt = 5 * 2* pi * r

= 10*pi*r

when r reaches 200feet, dA/dt = 10 * pi * 200

= 2000 pi (as per you answer..)