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 cut x that gives the largest volume.

A rectangular pen is fenced with 48 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 500 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.

An open box (no lid) with a square base must hold 13500 cm³. Use the least possible material, that is the smallest area of the base plus the four sides. Find the height h.

A small firm sells its product at a fixed market price of €50 per unit. Producing q units costs C(q) = 300 + 24q + 0.1q² euros. Find the maximum profit P.

A company sells q units per week. The price at which it can sell them is p = 60 − 1q euros per unit, and the cost is C(q) = 200 + 25q euros. Find the maximum weekly profit P.