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 largest total area A that can be enclosed.
An open box is made from a square sheet of cardboard 18 cm × 18 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 1000 cm³. The metal used is proportional to the total surface area. Find the radius r that uses the least material.
A rectangular plot of 200 m² is fenced on all four sides. The side along the street, of length x, costs €25 per metre; the other three sides cost €5 per metre. Find the length x of the street side that minimises the cost.
A theatre sells 1050 tickets per show at €20 each. Market research shows that every €1 increase in the ticket price means 25 fewer tickets sold (and every €1 decrease, 25 more). Find the maximum revenue R per show.
A theatre sells 1160 tickets per show at €25 each. Market research shows that every €1 increase in the ticket price means 40 fewer tickets sold (and every €1 decrease, 40 more). Find the maximum revenue R per show.
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