N2008 P1 Q7

A new flower-bed is being designed for a large garden. The flower-bed will occupy a rectangle \(x\) m by \(y\) m together with a semicircle of diameter \(x\) m, as shown in the diagram. A low wall will be built around the flower-bed. The time needed to build the wall will be 3 hours per metre for the straight parts and 9 hours per metre for the semicircular part. Given that a total time of 180 hours is taken to build the wall, find, using differentiation, the values of \(x\) and \(y\) which give a flower-bed of maximum area.[10]