An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed
region. Use these values to determine the maximum value of the objective function and the values of x and y for
which the maximum occurs.
-$$\begin{array} { l l }
\text { Objective Function } & \mathrm { z } = 23 \mathrm { x } + 19 \mathrm { y } \
\text { Constraints } & 0 \leq x \leq 10 \
& 0 \leq \mathrm { y } \leq 5 \
& 3 \mathrm { x } + 2 \mathrm { y } \geq 6
\end{array}$$
A)Maximum: 325; at (10, 5)
B)Maximum: 230; at (10, 0)
C)Maximum: 95; at (0, 5)
D)Maximum: 57; at (0, 3)

Question

An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed
region. Use these values to determine the maximum value of the objective function and the values of x and y for
which the maximum occurs.
-$$\begin{array} { l l } 
\text { Objective Function } & \mathrm { z } = 23 \mathrm { x } + 19 \mathrm { y } \
\text { Constraints } & 0 \leq x \leq 10 \
& 0 \leq \mathrm { y } \leq 5 \
& 3 \mathrm { x } + 2 \mathrm { y } \geq 6
\end{array}$$
A)Maximum: 325; at (10, 5)
B)Maximum: 230; at (10, 0)
C)Maximum: 95; at (0, 5)
D)Maximum: 57; at (0, 3)

Quizplus · Accepted Answer

To graph the system of inequalities, we can start by graphing the line 3x + 2y = 6, which is the same as y = -3/2x + 3. This line passes through the points (0,3) and (2,0), as shown below.

Next, we can shade the region above this line to satisfy the inequality 3x + 2y ≥ 6, and shade the feasible region where 0 ≤ x ≤ 10 and 0 ≤ y ≤ 5.

The corners of this feasible region are (0,0), (0,5), (2,2), (5,0), and (10,0). We can evaluate the objective function z = 23x + 19y at each of these corners to find the maximum value.

z(0,0) = 0
z(0,5) = 95
z(2,2) = 86
z(5,0) = 115
z(10,0) = 230

Therefore, the maximum value of the objective function is 325, which occurs at the corner (10,5).

An Objective Function and a System of Linear Inequalities Representing