2.25: Optimising the area of a window

0606/21/O/N/18 — Question 9 · 9 marks

2.25 diagram
In this question, all lengths are in metres.
The diagram shows a window formed by a semi-circle of radius rr on top of a rectangle with dimensions 2r2r by yy. The total perimeter of the window is 55.
(i)
Find yy in terms of rr.
[2]
(ii)
Show that the total area of the window is A=5r−πr22−2r2A = 5r - \dfrac{\pi r^{2}}{2} - 2r^{2}.
[2]
(iii)
Given that rr can vary, find the value of rr which gives a maximum area of the window and find this area. (You are not required to show that this area is a maximum.)
[5]