An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs. z=2x+3y {(x and y>=0),(2x+y<=8),(2x+3y<=12):}

Question
An objective function and a system of linear inequalities representing constraints are given.
a. Graph the system of inequalities representing the constraints.
b. Find the value of the objective function at each corner of the graphed region.
c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.
\(\displaystyle{z}={2}{x}+{3}{y}\)
\(\displaystyle{\left\lbrace\begin{array}{c} {x}{\quad\text{and}\quad}{y}\ge{0}\\{2}{x}+{y}\le{8}\\{2}{x}+{3}{y}\le{12}\end{array}\right.}\)

Answers (1)

2021-01-03
\(\displaystyle{x}\ge{0}\) (1)
\(\displaystyle{y}\ge{0}\) (2)
\(\displaystyle{2}{x}+{y}\le{8}\) (3)
\(\displaystyle{2}{x}+{3}{y}\le{12}\) (4)
a) Graph of given inequalities
b) The value at corner is given in graph.
c) find the maximum at which corner,
At point A (0,0)
z=2(0)+3(0) = 0
At point B (0,4)
z=2(0)+3(4) = 12
At point C (3,2)
z=2(3)+3(2) = 12
At point D (4,0)
z=2(4)+3(0) = 8
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