A rectangular pen is fenced with 144 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 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 largest possible volume V.
A rectangular plot of 600 m² is fenced on all four sides. The side along the street, of length x, costs €30 per metre; the other three sides cost €10 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 750 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.
A small firm sells its product at a fixed market price of €70 per unit. Producing q units costs C(q) = 5100 + 38q + 0.02q² euros. Find the output q that maximises profit.
A company sells q units per week. The price at which it can sell them is p = 180 − 2q euros per unit, and the cost is C(q) = 100 + 120q euros. Find the price p that maximises profit.
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