A company manufactures x units of Product A and y units of Product B, on two machines, I and II. It has been determined that the company will realize a profit of $2/unit of Product A and a profit of $7/unit of Product B. To manufacture a unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture a unit of Product B requires 9 min on Machine I and 4 min on Machine II. There are 5 hr of machine time available on Machine I and 3 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?
A company manufactures x units of Product A and y units of Product B, on two machines, I and II. It has been determined that the company will realize a profit of $2/unit of Product A and a profit of $7/unit of Product B. To manufacture a unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture a unit of Product B requires 9 min on Machine I and 4 min on Machine II. There are 5 hr of machine time available on Machine I and 3 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?
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What is the optimal profit? (Round your answer to the nearest whole number.)
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First of all let's consider the constraints/limits on the values for x and y:
- and since it doesn't make sense to produce a negative number of units.
- We are limited to 5 hours (or 300 minutes) on Machine I.
- To produce 1 unit of Product A it takes 6 minutes on Machine I. To produce x units of Product A it will take 6x minutes on machine I.
- To produce 1 unit of Product B it takes 9 minutes on Machine I. To produce y units of Product B it will take 9y minutes on machine I.
- Together we will need 6x + 9y minutes on Machine I to produce x units of Product A and y units of Product B.
- We are limited to 3 hours (or 180 minutes) on Machine II.
- To produce 1 unit of Product A it takes 5 minutes on Machine II. To produce x units of Product A it will take 5x minutes on machine II.
- To produce 1 unit of Product B it takes 4 minutes on Machine II. To produce y units of Product B it will take 4y minutes on machine II.
- Together we will need 5x + 4y minutes on Machine II to produce x units of Product A and y units of Product B.
So altogether the constraints on x and y are:
The profit from the sale of 1 unit is $2 for Product A and $7 for Product B. The profit from selling x units of Product A will be 2x dollars and the profit from selling y units will be 7y dollars. The total profit, then, will be:
Max P = 2x + 7y
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