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Rates of Change - Calculus Homework

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Hi everyone,

I'm struggling with my calculus homework on the rates of change. I have a number of problems I need to do, but I don't understand at all how to solve them. For the example below, could someone please walk me through the exact steps (and why you're doing them)?

A spark from a fire burns a hole in a paper napkin. The hole initially has a radius of 1cm and its area is increasing at a rate of 2cm²/s. Find the rate of change of the radius when the radius is 5 cm.

Thank you very much in advance for helping me out.

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First, let's write down the equation for the area of a circle: A = (pi)r2

The question asks for the rate of change of the radius. To find this, we must take the derivative of the area function with respect to time. This is because the rate of change of the area can be obtained from the rate of change of the radius and vice versa. Thus, we differentiate both sides (d/dt)

(da/dt) = (d/dt) (pi)r2

(da/dt) = (pi) 2r (dr/dt)        note: pi is a constant; the chain rule is applied. 2r = (dr/da) and we multiply that by (dr/dt) (we don't actually know the function for t in terms of r)

We know the value for (da/dt), it's just how much the area is changing with respect to time, which was given to us: 2 cm2/s.

2 = (pi) 2r (dr/dt)

Likewise, r was given to us as 5

2 = (pi) (2*5) (dr/dt)

2 = 10pi (dr/dt)

(2/(10pi)) = (dr/dt)

(dr/dt) = (1/5)pi

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