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A stone is dropped in a quiet lake and waves move in circles at the speed of 5 cm s-1. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

  1. 40 π cm2 s-1
  2. 80 π cm2 s-1
  3. 60 π cm2 s-1
  4. none of these

Answer (Detailed Solution Below)

Option 2 : 80 π cm2 s-1
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Detailed Solution

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CONCEPT :

Let f be a real-valued function which is a composite of two functions u and v; i.e., f = v o u. suppose t = u (x) and if both  and  exist.

Then differentiation of function (f) by chain rule is given by 

SOLUTION:

The area of a circle A with radius r is given by A = πr2 

∴ The rate of change of area (A) with respect to time (t) is given by chain rule is - 

It is given that :  cm

So when the radius of the circular wave will be 8 cm, the enclosed area will increase at the rate of 80π cm2/s.

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