Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 9 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.

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 closed cylindrical can (with a top and a bottom) must hold 500 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

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 680 tickets per show at €25 each. Market research shows that every €1 increase in the ticket price means 20 fewer tickets sold (and every €1 decrease, 20 more). Find the maximum revenue R per show.

A theatre sells 290 tickets per show at €20 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.