1.56: Optimising a Stadium-Shaped Lawn

Figure 2 shows a lawn $\displaystyle ABCDEF$, where $\displaystyle ABDE$ is a rectangle of length $\displaystyle y$ metres and width $\displaystyle 2x$ metres.

Each end of the lawn is in the shape of a semicircle of radius $\displaystyle x$ metres.

The perimeter of the lawn is $\displaystyle 90$ metres and the area of the lawn is $\displaystyle S\,\mathrm{m}^2$.

(a) Show that

where $\displaystyle k$ is a constant. State the value of $\displaystyle k$.

(b) Use calculus to find, to 4 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a maximum.

Justify that your value of $\displaystyle x$ gives the maximum value of $\displaystyle S$.

(c) Find, to the nearest whole number, the maximum value of $\displaystyle S$.