Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 18 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.

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 cut x that gives the largest volume.

A rectangular plot of 200 m² is fenced on all four sides. The side along the street, of length x, costs €15 per metre; the other three sides cost €5 per metre. Find the length x of the street side that minimises the cost.

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 radius r that uses the least material.

A company sells q units per week. The price at which it can sell them is p = 120 − 1q euros per unit, and the cost is C(q) = 1800 + 10q euros. Find the weekly output q that maximises profit.

A small firm sells its product at a fixed market price of €40 per unit. Producing q units costs C(q) = 2300 + 6q + 0.05q² euros. Find the output q that maximises profit.