A rectangular pen is fenced with 240 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.
An open box is made from a rectangular sheet of cardboard 14 cm × 30 cm by cutting equal squares of side x from the four corners and folding up the sides. Find the largest possible volume V.
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 radius r that uses the least material.
A closed cylindrical can (with a top and a bottom) must hold 1000 cm³. The metal used is proportional to the total surface area. Find the smallest possible surface area A.
A company sells q units per week. The price at which it can sell them is p = 60 − 0.1q euros per unit, and the cost is C(q) = 1600 + 20q euros. Find the weekly output q that maximises profit.
A small firm sells its product at a fixed market price of €50 per unit. Producing q units costs C(q) = 2900 + 16q + 0.02q² euros. Find the maximum profit P.
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