Optimization worksheet with answers

A farmer has 40 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the length y of the side parallel to the river for the largest area.

An open box is made from a rectangular sheet of cardboard 14 cm × 30 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the cut x that gives the largest volume.

A closed cylindrical can (with a top and a bottom) must hold 500 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

A rectangular plot of 600 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €10 per metre. Find the minimum total cost C.

A company sells q units per week. The price at which it can sell them is p = 110 − 2q euros per unit, and the cost is C(q) = 400 + 25q euros. Find the price p that maximises profit.

A company sells q units per week. The price at which it can sell them is p = 110 − 1q euros per unit, and the cost is C(q) = 300 + 30q euros. Find the price p that maximises profit.