Optimization worksheet with answers

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 farmer has 120 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 (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 smallest possible surface area A.

A theatre sells 600 tickets per show at €20 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 ticket price p that maximises revenue.

A small firm sells its product at a fixed market price of €60 per unit. Producing q units costs C(q) = 1000 + 28q + 0.05q² euros. Find the maximum profit P.