My brother asked me the following question:
A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
Well solving this using derivatives is not a difficult task. It turns out that,
"when the radius is 10 cm then the enclosed area is increasing at the rate of $80\pi cm^2/sec$".
Though he understands the mathematical working of the problem, he is putting forward the following question :
How and why are we finding the rate of change in area for a "fixed value" of the radius ?
I am at loss of words here, how do i explain this to him?