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 largest area A that can be enclosed.

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.

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the base side x.

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 height h of the can that uses the least material.

A theatre sells 490 tickets per show at €25 each. Market research shows that every €1 increase in the ticket price means 10 fewer tickets sold (and every €1 decrease, 10 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 = 160 − 1q euros per unit, and the cost is C(q) = 400 + 95q euros. Find the weekly output q that maximises profit.