Model and solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is either given or to be developed.

A rectangular storage container with a square base and an open top is to be constructed from sheet metal. The base costs $8 per square metre and the sides cost $5 per square metre. The container must have a volume of 32 cubic metres. (a) Show that the total cost C(x) as a function of the side length x metres of the square base is given by C(x) = 8x² + 640/x. [1 mark] (b) Find the side length of the base that minimises the total cost, and verify that this is indeed a minimum. [2 marks]

A rectangular garden is to be constructed against an existing wall. The gardener has 120 metres of fencing available and will use the wall as one side, so fencing is required for only three sides. If the length of the side parallel to the wall is \(x\) metres, which equation correctly models the area \(A(x)\) of the garden in factored form?

A farmer wishes to construct a rectangular enclosure adjacent to an existing straight fence. The enclosure will use the fence as one side (so only three sides need to be fenced). The farmer has 120 m of fencing material available. If the enclosure has length $x$ metres (perpendicular to the fence) and the side parallel to the fence has length $y$ metres, which expression correctly represents the optimisation model for maximising the area of the enclosure?

A water storage tank is being designed in the shape of a rectangular box with a square base. The tank must hold a volume of 8,000 litres. The base and top of the tank are made of reinforced concrete costing \$15 per square metre, while the four side walls are made of steel costing \$8 per square metre. (a) Express the total cost \(C\) of the tank's materials as a function of the side length \(x\) metres of the square base. (1 mark) (b) Find the value of \(x\) that minimises the total cost, and verify this is a minimum. (2 marks)