Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 10 cm × 16 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 is made from a square sheet of cardboard 24 cm × 24 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 €15 per metre; the other three sides cost €12 per metre. Find the minimum total cost C.

A closed cylindrical can (with a top and a bottom) must hold 1000 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

A theatre sells 2150 tickets per show at €25 each. Market research shows that every €1 increase in the ticket price means 50 fewer tickets sold (and every €1 decrease, 50 more). Find the maximum revenue R per show.

A small firm sells its product at a fixed market price of €65 per unit. Producing q units costs C(q) = 500 + 32q + 0.1q² euros. Find the output q that maximises profit.