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?

Question

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?

(x, y) =

What is the optimal profit? (Round your answer to the nearest whole number.)
$

Answer 1

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Answer 2

To find the optimal number of units for each product that will maximize the company's profit, we need to use linear programming. Linear programming is an optimization technique used to maximize or minimize a linear objective function subject to linear constraints.

Let's define our decision variables:
x = number of units of Product A
y = number of units of Product B

We want to maximize the company's profit, which is given by the objective function:
Profit = 2x + 7y

We need to consider the constraints given:
Machine I constraint: 6x + 9y ≤ 300 (5 hours of machine time on Machine I is equivalent to 300 minutes)
Machine II constraint: 5x + 4y ≤ 180 (3 hours of machine time on Machine II is equivalent to 180 minutes)

Additionally, since we cannot manufacture a negative number of units, we have:
x ≥ 0
y ≥ 0

Now, let's solve this linear programming problem to find the optimal solution.

Step 1: Set up the objective function and constraints:
Maximize: Profit = 2x + 7y
Subject to:
6x + 9y ≤ 300
5x + 4y ≤ 180
x ≥ 0
y ≥ 0

Step 2: Graph the feasible region by plotting the constraints on a graph. The feasible region is the area where all constraints are satisfied.

Step 3: Find the corner points of the feasible region. These are the vertices of the polygon formed by the feasible region.

Step 4: Evaluate the objective function at each corner point to find the maximum profit.

Alternatively, you can use a linear programming solver or software to solve this problem. By inputting the objective function and constraints, the solver will find the optimal solution for you.

Assuming you have solved this problem and found the optimal values of x and y, you can substitute them back into the objective function to find the optimal profit. Round the answer to the nearest whole number.