Answer
50 regular and 100 deluxe jet skis, and the profit is $\$35,000$.
Work Step by Step
Let us assume that x is the number of regular jet skis and y is the number of deluxe jet skis. Then, from the given information, the objective function is given below:
$ z=200x+250y $
And to maximize the profit, the constraints on the basis of the given information are as given below:
$\begin{align}
& x\ge 50 \\
& y\ge 75 \\
\end{align}$
And
$ x+y\le 150$
Use a graphical method to find the values of x and y.
Now, plot the graph for the inequalities with the feasible region.
Let us consider the corner points from the graph plotted $ A\left( 50,75 \right),B\left( 75,75 \right)$ and $ C\left( 50.100 \right)$.
Put the value of the x and y coordinates in the objective function for each corner point to compute the maximum profit as given below:
At corner point $ A\left( 50,75 \right)$,
$\begin{align}
& z=200\left( 50 \right)+250\left( 75 \right) \\
& =28,750
\end{align}$
At corner point $ B\left( 75,75 \right)$,
$\begin{align}
& z=200\left( 75 \right)+250\left( 75 \right) \\
& =33,750
\end{align}$
At corner point $ C\left( 50,100 \right)$,
$\begin{align}
& z=200\left( 50 \right)+250\left( 100 \right) \\
& =35,000
\end{align}$
The profit function is maximum at the corner point $ C\left( 50,100 \right)$.
Thus, the company needs to manufacture $50$ regular and $100$ deluxe jet skis to maximize the profit, and with the maximum profit of $\$35,000$.
