A Problem Of Optimization

Calculus Level 2

A company has got 90 $ 90\$ for x x objects of type A \mathrm{A} which cost 3 $ 3\$ each and y y objects of type B \mathrm{B} which cost 5 $ 5\$ each. Then, the product x y xy must be maximum. What is x + y x+y ?


The answer is 24.

Ervin Tronati by Ervin Tronati , 22, Italy

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1 solution

Marco Brezzi
Sep 10, 2017

By A M − G M AM-GM

3 x + 5 y ≥ 2 3 x ⋅ 5 y 3x+5y\geq 2\sqrt{3x\cdot 5y}

By definition 3 x + 5 y 3x+5y is 90 $ 90\$

90 ≥ 2 15 x y 90\geq 2\sqrt{15xy}

45 ≥ 15 x y 45\geq\sqrt{15xy}

Squaring both sides

4 5 2 ≥ 15 x y ⟺ x y ≤ 135 45^2\geq 15xy \iff xy\leq 135

Equality occurs when x = 15 x=15 and y = 9 y=9 , hence

x + y = 15 + 9 = 24 x+y=15+9=\boxed{24}

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