Competition problem — Algebra — Maths Olympiad Prep

Library / /194 of 520

Algebra Difficulty 5.1 AIME, harder Find the answer

11. A school is building a teaching building and needs 400 pieces of glass of size A and 500 pieces of glass of size B. The store has two types of glass, A and B. It is known that each piece of type A glass costs 48 yuan and can be cut into 4 pieces of size A and 6 pieces of size B; each piece of type B glass costs 58 yuan and can be cut into 5 pieces of size A and 5 pieces of size B. Therefore, the minimum cost for the school to purchase both types A and B glass is yuan.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

11.4720 .

Let the number of glass panels of specifications $A$ and $B$ to be purchased be $x$ and $y$ respectively. The constraints are
$\left\{\begin{array}{l} x \geqslant 0, y \geqslant 0, \\ 4 x+5 y \geqslant 400, \\ 6 x+5 y \geqslant 500 . \end{array}\right.$

The cost paid by the school $P=48 x+58 y$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.