A child throws a stone into a still lake causing a circular ripple to spread out

Question

A child throws a stone into a still lake causing a circular ripple to spread outward. If the radius of the ripple increases at a constant rate of 3/4 meters per second, how is the area of the disturbed water increasing when its radius is 15 meters? 6. For the function f(x) = x^3 + 3x^2 - 4, use your knowledge of derivatives to determine (a) the intervals on which f(x) is increasing/decreasing (b) the (z, y) coordinates of local extrema. Identify each as a maximum or minimum. c) the intervals of concavity and (x, y) coordinates of inflection points.

Explanation / Answer

Here we have given that rate of change in radius dr/dt = 3/4 m/sec

and dA/dr is asked here. Now we apply the chain rule here that is

dA/dt= dA/dr x dr/dt = 3/4(dA/dr)

Now as we have that area of circle A = pi r^2

Now differentiating it with respect to r, we get

dA/dr = pi d/dr(r^2) = 2 x r x pi

and when r= 5, then dA/dr = 2 x 3.14 x 15 = 94.2

So lastly using above chain rule

dA/dt = 94.2 x 3/4 = 94.2 x 0.75 = 70.65

So area is changing at the rate of 70.65 m^2/second.

This is the answer of question 5.

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