Rectangle In Rectangle

Erect similar right triangles of the length and width of the rectangle. Construct both sides of the circumscribing rectangle.

The area of the circumscribing rectangle is the product of the sides of the circumscribing rectangle.

$\text{area}=(l \cos (\theta )+w \sin (\theta )) (l \sin (\theta )+w \cos (\theta ))$

Compute the total derivative of the area treating the length and width as constants with respect to $\theta$ .

$(l \sin (\theta )+w \cos (\theta )) (w \cos (\theta )-l \sin (\theta ))+(l \cos (\theta )+w \sin (\theta )) (l \cos (\theta )-w \sin (\theta ))$

Solve for the derivative being 0, getting $\theta$ is $\frac{\pi}2 c_1+\frac{\pi }{4},c_1\in \mathbb{Z}$ .

Using the simplest value for $\theta$ and simplifying the area formula gives $\frac{1}{2} (l+w)^2$ .

Substituting the original values given gives the answer: 72.