2023 RI P1 Q11

One geometric optimization problem concerns finding the largest area axis-parallel rectangle inside an object.

The largest area rectangle problem has many industrial applications.For example, consider a piece of metal with a certain number of flaws in it.The piece of metal is cut into triangles and circles of equal area which does not contain any of the flaws.This problem can then be applied to find the maximum area of a rectangular sheet that can be contained in a circle and a triangle.

1. A rectangle with width \(2x\) units is inscribed inside a circle with centre \(O\) and radius \(r\)units as shown in the diagram below.
   
   1. Show that the area \(A\) units\(^{2}\) of the rectangle can be expressed as \(A=4x\sqrt{{{r}^{2}}-{{x}^{2}}}\).[1]
   2. Using differentiation, determine the area of the largest rectangle that can be inscribed in a circle of radius \(r\) units.[5]
   3. Hence state the area of the largest rectangle that can be inscribed in a circle of area \(\pi \) units\(^{2}\).[1]
2. The diagram below shows a triangle \(ABC\) of height \(1\) unit.
   
   Let \(y\) units be the height of the unshaded rectangle with \(2\) sides parallel to \(BC\). Let \(S\) denote the total areas of the three smaller shaded triangles and \(T\) denote the area of the triangle \(ABC\).
   1. Show that \(S=\left( {{y}^{2}}+{{\left( 1-y \right)}^{2}} \right)T\).[2]
   2. Determine the area of the largest rectangle that can be inscribed in a triangle of area \(\pi \) units\(^{2}\), showing your working clearly.[3]