finding the max of a function

Question

This question popped up during a nights drinking and it has been bugging me ever since. the conditions are: $$\left\{\begin{matrix} x+14y\leq 1820\\ x+25y\leq 2162.5\\ x\geq 0\\ y\geq 0 \end{matrix}\right.$$

The question is what is the maximum possible value of : \begin{matrix} \\ 4x+3y \end{matrix}.

It took us 3 lads around an hour and plenty of drawing to solve this.

My question is are there ways to solve this without plotting a graph and purely through equations.

I tried looking up the Lagrange multipliers method but that method falls flat for me as all the derivative are reduced to 1.

Answer 1

That region is a polyhedron whose sides are:

• the line segment that goes from $(0,0)$ to $(0,86.5)$;
• the line segment that goes from $(0,86.5)$ to $(1384.09,31.1364)$;
• the line segment that goes from $(1384.09,31.1364)$ to $(130,0)$;
• the line segment that goes from $(130,0)$ to $(0,0)$.

Since the gradient of $4x+3y$ is never $0$, the maximum has to be attained at one of the sides. Now, $(x,y)$ belongs to the first side if and only if $x=0$ and $y\in[0,86.5]$. So, $4x+3y$ attains the maximum ($259.5$) at $(0,86.5)$. Do the same thing for the other three sides, and you will get the answer.