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.
An open box is made from a rectangular sheet of cardboard 30 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.
An open box (no lid) with a square base must hold 108000 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 rectangular plot of 300 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €12 per metre. Find the minimum total cost C.
A small firm sells its product at a fixed market price of €60 per unit. Producing q units costs C(q) = 800 + 28q + 0.1q² 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 − 0.1q euros per unit, and the cost is C(q) = 1500 + 135q euros. Find the maximum weekly profit P.
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