Cheapest triangular garden

Problem statement

(2 marks) A garden in the shape of a right-angled triangle, with legs of 42 m and 56 m, is to be tiled. Inside the garden, a rectangular roofed area is to be marked out so that two of its sides are parallel to the legs of the triangle, one vertex coincides with the right-angle vertex of the triangle and the opposite vertex lies on its hypotenuse.

Tiling the roofed part costs €30/m² and the unroofed part €50/m², because those tiles have a special water-resistant treatment. Find the dimensions of the roofed part that make the cost of tiling the garden a minimum.

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Mathematics · Question 2. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.