Question

Tatiana manages a store that sells MP3 players. Each MX300 takes 4 hours to create the plastic pieces, 2 hours to create the electronics, and 19 hours of labor to assemble the final product. Each Rocker requires 2 hours to create the plastic pieces, 6 hours to create the electronics, and 14 hours of labor to assemble the final product. The factory has 440 hours to create the plastic pieces, 750 hours to create the electronics, and 2180 hours of labor to assemble the final product each week. If each MX300 generates $12 in profit, and each Rocker generates $14, how many of each of the MP3 players should Tatiana have the store sell each week to make the most profit?

Answer 1

To maximize profit, Tatiana needs to solve a system of inequalities using linear programming to find the optimal number of MX300 and Rocker MP3 players she should sell each week.

Step-by-step explanation:

To maximize the profit, Tatiana needs to determine the number of each MP3 player (MX300 and Rocker) to sell each week. Let's assume she sells x MX300 and y Rockers. The total time required to create the plastic pieces for the MX300 and Rocker is 4x + 2y hours. The total time required to create the electronics for the MX300 and Rocker is 2x + 6y hours. The total labor time required to assemble the MX300 and Rocker is 19x + 14y hours. From the given information, we have the following equations:

1. 4x + 2y ≤ 440 (plastic pieces time constraint)
2. 2x + 6y ≤ 750 (electronics time constraint)
3. 19x + 14y ≤ 2180 (labor time constraint)

The profit generated by selling x MX300 would be 12x dollars, and the profit generated by selling y Rockers would be 14y dollars. We want to maximize the profit, so we need to solve this system of inequalities and find the values of x and y that satisfy the constraints and maximize the profit function. The solution to this problem lies in linear programming, a mathematical technique used to optimize a function subject to a set of constraints.