Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 7 - Test - Page 879: 18

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 $\$35,000$.
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