Rectangle of maximum area with fixed perimeter

Statement

Among all rectangles with perimeter $24$, which one has maximum area?

Solution

Let $x$ and $y$ be the sides of the rectangle. From the fixed perimeter we obtain one side as a function of the other: $24 = 2x + 2y \implies y = 12 - x.$ The area, expressed as a function of $x$, is then $\text{Area} = x(12 - x) = -x^2 + 12x,$ that is, a parabola opening downwards: its maximum value occurs at the vertex. $x = -\frac{12}{2\cdot(-1)} = 6, \qquad y = 12 - 6 = 6.$ The rectangle with maximum area is therefore a square of side $6$, with area $\boxed{36}$.

Links

Topics: Equivalence and Pythagoras
Concepts: Area · Maximum · Parabola · Perimeter
Skills: Synthetic geometry · Modelling
Exercise type: Geometric problem · Modelling problem