Gce Model Questions Vol1 2023 Government Question 13

A local council has a total annual budget of £10 million to allocate among three services: Health (H), Education (E) and Infrastructure (I). Minimum allocations are £3 million for Health, £2 million for Education and £1 million for Infrastructure. The council's performance index is \(P = 2\sqrt{H} + 3\sqrt{E} + \sqrt{I}\), where the amounts are in millions of pounds.\n\n(a) Determine the allocation of the £10 million that maximises \(P\) subject to the minimum‑allocation constraints, and compute the maximum value of \(P\).\n(b) The government introduces a policy that raises the mandatory minimum for Health by £0.5 million (so the new minimum is £3.5 million) and, at the same time, increases the weight on Education by 10 % (so the term \(3\sqrt{E}\) becomes \(3.3\sqrt{E}\)). With the same total budget of £10 million, find the new optimal allocation and state whether the maximum performance index is higher, lower or the same as before.