Finding rate of change of area using derivatives 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?

It means that when the radius of the wave is $10cm$, in $1$ seconds time the area will be $80\pi cm^2$ bigger, i.e. $180\pi cm^2$.
