A company sells two products with information as follows: Products are made by machine. 4 units of Product A can be made with one machine hour and 2 units of Product B can be made with one machine hour. The company has a maximum of 3,000 machine hours available per month. The company can sell up to 7,000 units of Product A per month, and up to 3,000 units of Product B for the month. What is the maximum amount of contribution margin that the company could earn in a month given the stated constraints?

Question

A company sells two products with information as follows:  Products are made by machine. 4 units of Product A can be made with one machine hour and 2 units of Product B can be made with one machine hour. The company has a maximum of 3,000 machine hours available per month. The company can sell up to 7,000 units of Product A per month, and up to 3,000 units of Product B for the month. What is the maximum amount of contribution margin that the company could earn in a month given the stated constraints?

Quizplus · Accepted Answer

Let's define:
- x: number of machine hours used for Product A
- y: number of machine hours used for Product B

Our objective is to maximize the contribution margin, which is given by:
Contribution Margin = (Price per unit * Units sold) - (Variable cost per unit * Units sold)

We need to calculate the contribution margin for each product first:
- Product A:
Contribution Margin A = (30 * Units sold A) - (20 * Units sold A) = 10 * Units sold A
We know that the company can sell up to 7,000 units of Product A per month, so the maximum contribution margin for Product A is:
Contribution Margin A max = 10 * 7,000 = 70,000

- Product B:
Contribution Margin B = (50 * Units sold B) - (30 * Units sold B) = 20 * Units sold B
We know that the company can sell up to 3,000 units of Product B per month, so the maximum contribution margin for Product B is:
Contribution Margin B max = 20 * 3,000 = 60,000

Now, let's write the constraints:
- 4x + 2y ≤ 3,000 (total hours available)
- x ≤ 7,000 (maximum units of Product A)
- y ≤ 3,000 (maximum units of Product B)
We can rewrite the first constraint as:
2x + y ≤ 1,500

Let's graph these constraints and identify the feasible region:

The feasible region is the shaded area. We want to maximize the contribution margin, so we need to evaluate this function at each of the corner points of the feasible region:

Corner point 1: x = 0, y = 0
Contribution Margin = 0

Corner point 2: x = 0, y = 1,500
Contribution Margin = 20 * 1,500 = 30,000

Corner point 3: x = 500, y = 1,000
Contribution Margin = (30 * 500) - (20 * 500) + (50 * 1,000) - (30 * 1,000) = 36,000

Corner point 4: x = 1,750, y = 0
Contribution Margin = (30 * 1,750) - (20 * 1,750) = 17,500

Therefore, the maximum contribution margin that the company could earn in a month is $36,000, which corresponds to the corner point (500, 1,000) and the production of 2,000 units of Product A and 1,000 units of Product B.

A Company Sells Two Products with Information as Follows