Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 30 cm × 48 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 120 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 length x that gives the largest total area.

A closed cylindrical can (with a top and a bottom) must hold 355 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.

A rectangular plot of 450 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €5 per metre. Find the length x of the street side that minimises the cost.

A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 3500 + 22q + 0.1q² euros. Find the maximum profit P.

A theatre sells 490 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.