Related Rates: Is the Derivative of a Changing Circle's Area Constant?

[rook_b]

Homework Statement

Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is dA/dt constant? Explain.

Homework Equations

A = 2(pi)r^2
dA/dt = 2(pi)(r)*dr/dt

The Attempt at a Solution

I can't decide. You see if dr/dt is constant and multiplied by 2(pi) then dA/dt must be constant. But, r, what is r? It must be that 'r' is a function of time, I think. Or, r is an initial value that grows as the rate dr/dt. A constant rate times a constant should yield a constant rate, but multiplied by a variable it will yield a variable rate. I hate r.

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[HallsofIvy]

If dr/dt is any constant other than 0, yes, r is a variable and so dA/dt is a variable. Don't hate r- it's harmless.

 

[rook_b]

Ah, thank you once again HallsofIvy.

 

[rook_b]

Hmm, one more question. Where does time t come into this? Specifically, it seems that r must be a function of time t so is r really r(t). That is, if I wanted to graph this as time vs dA/dt. A short confirmation by anyone would be appreciated.

 

[brh2113]

Yes, r is a function of t. dr/dt is the derivative of that function with respect to t. Since dr/dt does not equal 0, r(t) is not constant. We know that dr/dt does not equal 0 because the problem says that the radius of the circle is changing with respect to time.