If the constraints are $x+y\le 10$, $x\ge 2$, $y\ge 3$, what is the maximum value of $Z=5x+2y$?

Solution

1. The feasible region is defined by the constraints $x+y\le 10$, $x\ge 2$, and $y\ge 3$, which form a triangle with vertices at $(2,3)$, $(7,3)$, and $(2,8)$.
2. Evaluate the objective function $Z=5x+2y$ at each vertex: $$\begin{array}{rl}Z(2,3)& =5(2)+2(3)\\ & =16\end{array}$$, $$\begin{array}{rl}Z(7,3)& =5(7)+2(3)\\ & =41\end{array}$$, and $$\begin{array}{rl}Z(2,8)& =5(2)+2(8)\\ & =26\end{array}$$.
3. Comparing these values, the maximum value is $41$, which corresponds to the closest option provided.
4. Hence the answer is (B).