Optimization worksheet with answers

An open box is made from a square sheet of cardboard 48 cm × 48 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.

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 x of each side perpendicular to the river for the largest area.

An open box (no lid) with a square base must hold 62500 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 250 cm³. The metal used is proportional to the total surface area. Find the height h of the can that uses the least material.

A small firm sells its product at a fixed market price of €55 per unit. Producing q units costs C(q) = 1200 + 46q + 0.01q² euros. Find the maximum profit P.

A theatre sells 290 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 ticket price p that maximises revenue.