Question

A manufacturer of wheelbarrows makes two models, Deluxe and Standard. For the deluxe model, he needs machine A for 2 minutes and machine B for 3 minutes. For the standard model, he needs machine A for 3 minutes and machine B for 2 minutes. Both machine A and B are available for at most 48 minutes. He knows from experience; he will sell standard model at least twice as many as the deluxe model. The deluxe earns him a profit 25 and standard model earns 20. How many of each model be made to maximize the profit?

Answer 1

To maximize profits for manufacturing Deluxe and Standard wheelbarrow models, we must solve a linear programming problem with the objective function P = 25x + 20y and subject to constraints from machine time availability and the requirement to produce at least twice as many Standard models as the Deluxe.

Step-by-step explanation:

The student is tasked with solving a linear programming problem to maximize profits for a manufacturer of wheelbarrows that makes Deluxe and Standard models. We are looking to establish how many of each model should be made under machine time constraints and the condition that at least twice as many Standard models as the Deluxe ones are sold.

To maximize the profit, we need to formulate the objective function and the constraints:

• Let x be the number of Deluxe models.
• Let y be the number of Standard models.

The objective function representing total profit, P, in terms of x and y will be:

P = 25x + 20y

The constraints based on machine availability are:

1. 2x + 3y ≤ 48 (Machine A availability)
2. 3x + 2y ≤ 48 (Machine B availability)
3. y ≥ 2x (Standard model at least twice as many as Deluxe)
4. x ≥ 0, y ≥ 0 (Non-negativity constraints)

The student should then use graphical methods or linear programming techniques such as the Simplex method to find the values of x and y that maximize P subject to these constraints.