DHS Tutorial Q2

A pond with a constant depth of \(1\)m is being designed for a park. The pond comprises a rectangle \(x\)m by \(y\)m and two semicircles of diameter \(y\)m, as shown in the diagram. The cost to build a boundary around the pond is \(\$30\) per metre for the straight parts and \(\$60\)per metre for the semi-circular parts. Given that the budget for the boundary is fixed at \(\$6000\), using differentiation or otherwise, find in terms of \(\pi \), the exact values of \(x\) and \(y\) which give the pond a maximum volume.[7]