A rectangular pen is fenced with 96 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.
An open box is made from a square sheet of cardboard 12 cm × 12 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 smallest possible surface area A.
A rectangular plot of 1200 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 minimum total cost C.
A company sells q units per week. The price at which it can sell them is p = 80 − 0.5q euros per unit, and the cost is C(q) = 200 + 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 = 80 − 0.25q euros per unit, and the cost is C(q) = 1700 + 15q euros. Find the price p that maximises profit.
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