Optimization worksheet with answers

A farmer has 320 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 square sheet of cardboard 18 cm × 18 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 rectangular plot of 400 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €8 per metre. Find the minimum total cost C.

An open box (no lid) with a square base must hold 500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

A company sells q units per week. The price at which it can sell them is p = 110 − 0.25q euros per unit, and the cost is C(q) = 500 + 70q euros. Find the maximum weekly profit P.

A company sells q units per week. The price at which it can sell them is p = 70 − 0.2q euros per unit, and the cost is C(q) = 1000 + 25q euros. Find the weekly output q that maximises profit.