Question #40554

A factory produces three chemicals $A, B,$ and $C$ in quantities $x, y,$ and $z$ (in kg) respectively, where $x, y, z \ge 0$. The total production must be exactly $12$ kg. Due to safety regulations, the production state $(x,y,z)$ must satisfy a stability constraint: its perpendicular distance from the plane $x - 2y + 2z = 0$ cannot exceed $2$ units. The profit per kg for chemicals $B$ and $C$ are $4$ and $3$ respectively. The profit per kg for chemical $A$ is $\lambda$, where $\lambda$ is the maximum possible value of $|z_1 - z_2|$ for complex numbers $z_1, z_2$ satisfying $|z_1| = 2$ and $|z_2 - 3 - 4i| = 1$. Find the maximum total profit the factory can achieve.