We have been given a system of inequalities and an objective function.
The inequalities are given as:
[tex]y\leq 2x\\
x+y\leq 45\\
x\leq 30\\[/tex]
And the objective function is given as:
[tex]P=25x+20y[/tex]
In order to find the minimum value of the objective function at the given feasible region, we need to first graph the region.
The graph of the region is shown below:
From the graph, we can see that corner points of the feasible region are:
(x,y) = (15,30),(30,15) and (30,60).
Now we will evaluate the value of the objective function at each of these corner points and then we will compare which of those values is minimum.
[tex]\text{At (15,30)}\Leftrightarrow P=25\cdot 15+20\cdot 30=975\\
\text{At (30,15)}\Leftrightarrow P=25\cdot 30+20\cdot 15=1050\\
\text{At (30,60)}\Leftrightarrow P=25\cdot 30+20\cdot 60=1950\\[/tex]
Hence the minimum value of objective function is 975 and it occurs at x = 15 and y = 30
