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.
A farmer has 80 m of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. Find the length y of the side parallel to the river for the largest area.
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 rectangular plot of 450 m² is fenced on all four sides. The side along the street, of length x, costs €40 per metre; the other three sides cost €5 per metre. Find the minimum total cost C.
A theatre sells 780 tickets per show at €25 each. Market research shows that every €1 increase in the ticket price means 20 fewer tickets sold (and every €1 decrease, 20 more). Find the maximum revenue R per show.
A small firm sells its product at a fixed market price of €25 per unit. Producing q units costs C(q) = 900 + 6q + 0.05q² euros. Find the maximum profit P.
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