suppose we have following question
A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of $x$, then what is the width of the path in terms of $x$?
so we have following picture right
because we dont know if path of uniform width is square or not how can i find width?if area of square is $x$,then length is $\sqrt{x}$,but what about second figure?suppose it's length are $a$ and $b$,then $a*b=x$,then how can i continue?
Best Answer
You would have
$$4 w \sqrt{x} + 4 w^2 = x$$
(i.e., four rectangles + four squares that make up path, width of path = $w$)
Solve for $w$:
$$2 w = \frac{-\sqrt{x} \pm \sqrt{2 x}}{2}$$
Choose the positive solution:
$$w=\frac{\sqrt{2}-1}{4} \sqrt{x}$$