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 largest area A that can be enclosed.

An open box is made from a square sheet of cardboard 30 cm × 30 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

An open box (no lid) with a square base must hold 4000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

A closed cylindrical can (with a top and a bottom) must hold 330 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

A theatre sells 950 tickets per show at €12 each. Market research shows that every €1 increase in the ticket price means 25 fewer tickets sold (and every €1 decrease, 25 more). Find the maximum revenue R per show.

A company sells q units per week. The price at which it can sell them is p = 80 − 0.5q euros per unit, and the cost is C(q) = 800 + 15q euros. Find the price p that maximises profit.