2026 CCHY P2 Q10

A company designs an open-top rectangular water tank with a square base of side \(x\) metres and height \(h\) metres for the four walls. The tank must hold exactly \(256\text{ m}^3\) of water. The cost of materials is \(\$90\) per m\(^2\) for the base and \(\$40\) per m\(^2\) for the four walls. After construction, the outside of the four walls is painted. The paint costs \(\$15\) per m\(^2\).

1. Show that the material cost \(C\) dollars can be written as \(C=90x^2+\dfrac{40960}{x}\).[2]
2. Find the value of \(x\) for which the material cost is a minimum.[5]
3. Hence, find the minimum cost, giving your answer to the nearest dollar.[1]
4. Show that the cost of paint, \(P\), dollars is given by \(P=\dfrac{15360}{x}\).[2]
5. Given that the side length of the square base cannot exceed 8 m, determine the dimensions of the tank for which the cost of paint job is the least.[1]