Optimization worksheet with answers

An open box is made from a square sheet of cardboard 42 cm × 42 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 pen is fenced with 360 m of fencing and divided into two parts by one extra fence parallel to the sides of length x (three fences of length x and two of length y in total). Find the largest total area A that can be enclosed.

A closed cylindrical can (with a top and a bottom) must hold 400 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

A rectangular plot of 800 m² is fenced on all four sides. The side along the street, of length x, costs €15 per metre; the other three sides cost €10 per metre. Find the minimum total cost C.

A theatre sells 380 tickets per show at €15 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 440 tickets per show at €15 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.