Solve the following linear programming problem. Restrict $x \geq 0$ and $y \geq 0$. \[ \begin{array}{l} \text { Minimize } g=40 x+70 y \text { subject to the following. } \\ 11 x+15 y \geq 225 \\ x+3 y \geq 27 \\ (x, y)=(\square) \\ g= \\ \end{array} \]
Solution
Step 1 ：This is a linear programming problem. We are asked to minimize the function \(g=40x+70y\) subject to the constraints \(11x+15y \geq 225\) and \(x+3y \geq 27\). We also have the non-negativity constraints \(x \geq 0\) and \(y \geq 0\).
Step 2 ：To solve this problem, we can use a method for solving linear programming problems that minimizes a linear objective function subject to linear equality and inequality constraints.
Step 3 ：The optimal value of the function \(g=40x+70y\) is 880, and this is achieved when \(x=15\) and \(y=4\). These values satisfy all the constraints of the problem.
Step 4 ：Final Answer: The optimal solution is \((x, y) = (15, 4)\) and the minimum value of \(g\) is \(880\). So, \((x, y) = \boxed{(15, 4)}\) and \(g = \boxed{880}\).
