Of all rectangles of area 100, which has the smallest perimeter?

The correct answer is 10

Let $x$ denote one side of the rectangle.

As the area of the rectangle is 100, the other side will be $\dfrac{100}{x}$

Perimeter of the rectangle = $f(x)$ = $2x$ + $2\dfrac{100}{x}$

$f '(x)$ = $2 - \dfrac{200}{x^2}$



Let us solve this equation for $f '(x) = 0$

$2 - \dfrac{200}{x^2} = 0$ ⇒ $x = ± 10$

As $x$ represents the side of a rectangle, it cannot be negative.

Hence a rectangle with area 100 and side 10 will have the smallest perimeter.