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?

• Try googling "Zeno arrow paradox". – David Mitra Jul 24 '15 at 8:11
• Because you are trying to evaluate the rate of increase of area at a particular point in time when the radius has the value 10 – David Quinn Jul 24 '15 at 9:01

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. $180\pi cm^2$.