Certainly! Let's walk through the steps to determine the maximum revenue given the vertices of the feasible region and the optimization equation.
We are given the vertices of the feasible region:
[tex]\[
(0, 0), (0, 40), (40, 30), (100, 0)
\][/tex]
The optimization equation is given as:
[tex]\[
5b + 7.5n
\][/tex]
Now, we will substitute each ordered pair into this equation to determine the revenue generated at each vertex:
1. Vertex (0, 0):
[tex]\[
5(0) + 7.5(0) = 0
\][/tex]
Revenue: \[tex]$0.00
2. Vertex (0, 40):
\[
5(0) + 7.5(40) = 0 + 300 = 300
\]
Revenue: \$[/tex]300.00
3. Vertex (40, 30):
[tex]\[
5(40) + 7.5(30) = 200 + 225 = 425
\][/tex]
Revenue: \[tex]$425.00
4. Vertex (100, 0):
\[
5(100) + 7.5(0) = 500 + 0 = 500
\]
Revenue: \$[/tex]500.00
To summarize:
- At vertex [tex]\((0, 0)\)[/tex], the revenue is \[tex]$0.00
- At vertex \((0, 40)\), the revenue is \$[/tex]300.00
- At vertex [tex]\((40, 30)\)[/tex], the revenue is \[tex]$425.00
- At vertex \((100, 0)\), the revenue is \$[/tex]500.00
The maximum revenue occurs at vertex [tex]\((100, 0)\)[/tex] with a revenue of \$500.00.