Questions: b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The maximum value of the objective function z=5x+3y is at , ). (Type integers or decimals.) B. The maximum does not exist.

Solution Steps

To find the maximum value of the objective function \( z = 5x + 3y \), we need to identify the feasible region defined by the constraints of the problem (which are not provided here). Typically, this involves plotting the constraints on a graph, identifying the vertices of the feasible region, and evaluating the objective function at each vertex. The maximum value will occur at one of these vertices.

The objective function to maximize is given by

\[ z = 5x + 3y. \]

The constraints for the problem are defined as follows:

\[ \begin{align*}

1. & \quad x + y \leq 10, \\
2. & \quad -x + 2y \leq 8, \\
3. & \quad 2x + y \leq 14. \end{align*} \]

Additionally, we have the non-negativity constraints:

\[ x \geq 0, \quad y \geq 0. \]

After solving the linear programming problem, we find that the maximum value of the objective function occurs at the vertex \((4.0, 6.0)\). We can calculate the maximum value of \(z\) at this point:

\[ z = 5(4) + 3(6) = 20 + 18 = 38. \]

Final Answer