A stone is dropped into a still pond. Concentric circular ripples spread out, and the radius of the disturbed region increases at the rate of 16 cm/sec.

Question

A stone is dropped into a still pond. Concentric circular ripples spread out, and the radius of the disturbed region increases at the rate of 16 cm/sec. At what rate does the area of the disturbed region increase when its radius is 4 cm?

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Defined variables-                                                                           

Numerical facts known about the variables and their

           derivatives with respect to t-                                                

The unknown (what you wish to find)-                                           

The equation to relate the variable that depend on t-                       

Final answer (includes the solution on differentiation)-                   

Interpretation-              

Answer 1

The three variables that we're working with are:

A = area of the circle

r = radius of the circle

t = time in seconds

Since the radius is increasing at a rate of 16cm/sec, we can write r in terms of t with:

r = 16t

Now we want to know the rate of change of the area at a certain radius. We can substitute the radius in terms of time and take the derivative to calculate a formula for rate of change:

A = πr²

A = π(16t)²

A = 256πt²

A' = 512πt

Now all we have to do is find out what value of t will make it so that the radius is at 4cm and use that to find the rate of change (A') at that value. Relatively simple:

4 = 16t

A'(¹/₄) = 128π