Maximize $p=x+2 y$ subject to
\[
\begin{array}{l}
x+5 y \leq 18 \\
4 x+y \leq 15 \\
x \geq 0, y \geq 0
\end{array}
\]
Answer
The final answer is: The maximum value of the objective function \(p=x+2y\) subject to the constraints \(x+5y\leq18\), \(4x+y\leq15\), and \(x\geq0\), \(y\geq0\) is \(\boxed{9}\) at the point \((x, y) = \boxed{(3, 3)}\).
Step 1 ：Define the objective function as \(p=x+2y\).
Step 2 ：Set the constraints as \(x+5y\leq18\), \(4x+y\leq15\), \(x\geq0\), and \(y\geq0\).
Step 3 ：Use linear programming to solve the problem. The objective is to maximize the objective function subject to the constraints.
Step 4 ：The optimal values for \(x\) and \(y\) are both 3.0.
Step 5 ：The maximum value of the objective function is 9.0.
Step 6 ：The final answer is: The maximum value of the objective function \(p=x+2y\) subject to the constraints \(x+5y\leq18\), \(4x+y\leq15\), and \(x\geq0\), \(y\geq0\) is \(\boxed{9}\) at the point \((x, y) = \boxed{(3, 3)}\).
