Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!

Let there be $n$ equal circles of radius $r$ arranged in a row inside a rectangle.

Then:

Rectangle height = $2r$

Rectangle width = $2nr$

Therefore,

Area of rectangle = $(2r)(2nr) = 4nr^2$

Total area of the $\pi$ circles:

$$n(\pi r^2) = n\pi r^2$$

Hence, $$\frac{\text{Area occupied by circles}}{\text{Area of rectangle}} = \frac{n\pi r^{2}}{4nr^{2}} = \frac{\pi}{4}$$

Thus, irrespective of the number of circles,

Area occupied by circles = $\frac{\pi}{4}$ of the rectangle

Hence, proved.