Question

A manufacturer produces two types of machines from parts x1 and x2. the resources needed forproducing machines x1 and x2 and the corresponding profits are in the table below:

machine labor hrs/unit capital ($/unit) profit ($)
x1 10 40 40
x2 20 30 50

there are 400 hrs of labor and $1200 worth of capital available per day for assembling themachines. how many machines, x1 and x2, should be produced per day to maximize profit?

Answer 1

To maximize profit, the manufacturer needs to solve a linear programming problem using the given labor and capital constraints to find the optimal mix of machines X1 and X2.

Step-by-step explanation:

To maximize profit, the manufacturer should determine the number of machines X1 and X2 that can be produced given the constraints of labor hours and capital available. This problem can be approached using linear programming, where the objective function to maximize is the profit, and the constraints are the available labor hours and capital.

Let's denote the number of machines X1 produced as x and machines X2 as y. The objective function for profit (P) will thus be:

P = 40x + 50y

The constraints based on labor hours and capital, respectively, are:

• 10x + 20y ≤ 400
• 40x + 30y ≤ 1200

To solve this problem, construct a graph with x and y axes representing the number of machines X1 and X2, respectively. Plot the constraints and look for the feasible region, then identify the corner points of this region. Evaluate the profit function at each corner point to determine which one provides the maximum profit.

This optimization problem usually results in a mix of both machines X1 and X2 to make the best use of the available resources. The exact numbers will be determined by the intersection points of the constraints and the highest profit point on the feasible region's boundary.