Question #41700

Let $A$ be the origin and $B$ be the point representing the complex number $3+4i$ on the Argand plane. Let $C$ be a variable point representing the complex number $w$ such that $|w - (8+6i)| \le 2$. Let $M$ and $m$ be the maximum and minimum possible areas of $\triangle ABC$, respectively.

A factory produces quantities $x$ and $y$ (where $x, y \in \mathbb{R}$) of two goods, X and Y. The profit per unit of X is $M$ and per unit of Y is $m$. The production constraints are:

1. The total quantity produced cannot exceed 10 units.
2. The quantity of X produced must be at least the quantity of Y.
3. At least 2 units of X and 1 unit of Y must be produced.

What is the maximum possible profit?