A factory can produce two products, x and y, with a profit approximated by P = 14x + 22y – 900. The production of y can exceed x by no more than 200 units. Moreover, production levels are limited by the formula x + 2y ≤ 1600. What production levels yield maximum profit? (1 point)
x = 400; y = 600
x = 0; y = 0
x = 1,600; y = 0
x = 0; y = 200
x = 400; y = 600
To find the production levels that yield maximum profit, we need to solve the given constraints:
1) The production of y can exceed x by no more than 200 units.
2) Production levels are limited by the formula x + 2y ≤ 1600.
To solve this problem, we can use the method of linear programming.
Step 1: Convert the inequality constraint into an equation:
x + 2y = 1600
Step 2: Plot the graph of the equation:
The graph will be a straight line passing through the points (0, 800) and (1600, 0).
Step 3: Plot the feasible region:
To find the feasible region, we need to consider the given constraint that the production of y can exceed x by no more than 200 units. This means y - x ≤ 200.
Plotting this line on the graph will give us a region where y is greater than or equal to x and y - x ≤ 200.
Step 4: Identify the corner points of the feasible region:
The corner points of the feasible region will occur on the intersection of the two lines.
By inspecting the graph, we can see that the corner points are:
(0, 0), (0, 200), (400, 600), and (1600, 0).
Step 5: Calculate the profit for each corner point:
Plug the values of x and y for each corner point into the profit equation P = 14x + 22y - 900 to calculate the profit at each point.
The profit at each corner point is:
(0, 0) = 14(0) + 22(0) - 900 = -900
(0, 200) = 14(0) + 22(200) - 900 = 3,500
(400, 600) = 14(400) + 22(600) - 900 = 15,100
(1600, 0) = 14(1600) + 22(0) - 900 = 20,900
Step 6: Compare the profit at each corner point:
The maximum profit is achieved at the corner point with the highest profit.
By comparing the profit at each corner point, we can see that the production levels that yield the maximum profit are:
x = 400, y = 600
Therefore, the correct answer is:
x = 400; y = 600
To find the production levels that yield maximum profit, we need to maximize the profit function P = 14x + 22y - 900, subject to the given constraints.
First, let's examine the constraints. The first constraint states that the production of y can exceed x by no more than 200 units:
y - x ≤ 200
Rearranging this inequality, we get:
y ≤ x + 200
The second constraint is given by the formula:
x + 2y ≤ 1600
Now, we can solve this optimization problem using the method of linear programming.
Step 1: Graph the feasible region:
To graph the feasible region, we plot the equations x + 2y ≤ 1600 and y ≤ x + 200 on a coordinate plane.
Step 2: Identify the vertices of the feasible region:
The vertices of the feasible region are the points where the lines intersect or where they touch the boundary line.
Step 3: Substitute the vertices into the profit function:
For each vertex, substitute the x and y values into the profit function P = 14x + 22y - 900 to find the profit.
Step 4: Determine the maximum profit:
Compare the profits obtained from each vertex and identify the highest value. This will give us the production levels that yield the maximum profit.
By following these steps, we can determine the production levels that yield the maximum profit. Unfortunately, without a visual representation of the graph or more information, we cannot calculate the exact values for x and y.