2025 RVHS Promo Q4

A running track is made up of a rectangle and 2 semicircles, as shown in the diagram below. The running track has a total perimeter of 400 m, and the width and length of the rectangular part is \(2x\) m and \(y\) m respectively.

1. Show that area of the running track, \(A\) m\(^2\), is given by \(A=400x-\pi x^2\).[3]
2. Use differentiation to find the maximum value of \(A\) in terms of \(\pi\), proving that it is a maximum.[4]