Optimization worksheet with answers

An open box is made from a rectangular sheet of cardboard 15 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 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 rectangular plot of 450 m² is fenced on all four sides. The side along the street, of length x, costs €20 per metre; the other three sides cost €8 per metre. Find the minimum total cost C.

An open box (no lid) with a square base must hold 32000 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the minimum surface area S.

A company sells q units per week. The price at which it can sell them is p = 110 − 0.5q euros per unit, and the cost is C(q) = 600 + 45q euros. Find the maximum weekly profit P.

A company sells q units per week. The price at which it can sell them is p = 180 − 0.25q euros per unit, and the cost is C(q) = 400 + 135q euros. Find the weekly output q that maximises profit.