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the radius of a circular oil slick expands at a rate of 5 m/min. (a) how fast is the area of the oil slick increasing when the radius is 20 m? (b) if the radius is 0 at time , how fast is the area increasing after 5 mins?

User Krisk
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a) To solve this problem, we need to use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius. We need to find how fast the area is increasing, or dA/dt, when the radius is 20 m and the rate of change of the radius is 5 m/min.

First, we can differentiate the formula for the area of a circle with respect to time to get the rate of change of the area:

dA/dt = 2πr(dr/dt)

Substituting the given values, we get:

dA/dt = 2π(20)(5) = 200π

Therefore, the area of the oil slick is increasing at a rate of 200π square meters per minute when the radius is 20 m.

b) If the radius is 0 at time t = 0, then at time t = 5 mins, the radius will be:

r = 5t = 5(5) = 25 m

Using the same formula as in part a), we can find how fast the area is increasing:

dA/dt = 2πr(dr/dt) = 2π(25)(5) = 250π

Therefore, the area of the oil slick is increasing at a rate of 250π square meters per minute after 5 minutes.
User Meistermeier
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