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)
A. x = 400; y = 600
B. x = 0; y = 0
C. x = 1,600; y = 0
D. x = 0; y = 200
To find the production levels that yield maximum profit, we need to optimize the profit function while considering the given constraints.
Let's begin by analyzing the constraints provided:
1) The production of y cannot exceed x by more than 200 units.
y - x ≤ 200
2) The total production level is limited by the formula: x + 2y ≤ 1600.
Now, let's consider the profit function:
P = 14x + 22y - 900
To optimize this function, we can use the method of linear programming.
Step 1: Convert the inequalities to equations:
1) y - x = 200
2) x + 2y = 1600
Step 2: Solve the system of equations:
From equation 2) we can solve for x:
x = 1600 - 2y
Substitute this expression for x in equation 1):
y - (1600-2y) = 200
3y = 1800
y = 600
Substitute the value of y back into equation 2):
x + 2(600) = 1600
x + 1200 = 1600
x = 400
Therefore, the production levels that yield maximum profit are:
x = 400
y = 600
So, the answer is:
A. x = 400; y = 600
To find the production levels that yield the maximum profit, we need to optimize the profit function while considering the given constraints.
First, we need to determine the feasible region, which is defined by the constraint x + 2y ≤ 1600. We can rewrite this constraint as y ≤ (1600 - x)/2.
Next, we need to consider the constraint that the production of y can exceed x by no more than 200 units. We can express this constraint as y - x ≤ 200, or rearranging, x ≥ y - 200.
To visualize the feasible region, we can plot the two constraints on a graph:
x + 2y ≤ 1600
y ≤ (1600 - x)/2
x ≥ y - 200
Let's plot these constraints:
First, we plot the line x + 2y = 1600:
To graph this line, we can find the x and y-intercepts by setting x and y to 0:
At x = 0, we have 2y = 1600, y = 800. So one point on the line is (0, 800).
At y = 0, we have x = 1600. So another point on the line is (1600, 0).
Plotting these points and drawing the line x + 2y = 1600, we can see that it intersects the x-axis at x = 1600 and the y-axis at y = 800.
Next, we plot the line x = y - 200:
To graph this line, we can find the x and y-intercepts by setting x and y to 0:
At x = 0, we have y = 200. So one point on the line is (0, 200).
At y = 0, we have x = -200. Since x cannot be negative in this context, this point is not included in the feasible region.
Plotting the point (0, 200) and drawing the line x = y - 200, we get a vertical line passing through (0, 200).
The feasible region is the region bounded by these two lines and the x-axis.
To find the production levels that yield maximum profit, we need to optimize the profit function subject to the given constraints.
Let's analyze the given information:
- The profit function is P = 14x + 22y - 900.
- The production of y can exceed x by no more than 200 units, which can be written as y - x ≤ 200.
- The 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 given constraints into a standard form.
- Convert y - x ≤ 200 to x - y ≥ -200.
- Rewrite x + 2y ≤ 1600 as -x - 2y ≥ -1600.
Step 2: Plot the feasible region by graphing the inequalities.
- Start by graphing the line x - y = -200.
- Find two points on this line by setting x = 0 and solving for y, and setting y = 0 and solving for x.
- Connect these points to draw the line.
- Choose a side of the line to represent the feasible region.
- Graph the line -x - 2y = -1600 in the same manner.
- The feasible region is the area bounded by these lines and the axes.
Step 3: Identify the corner points of the feasible region.
- The corner points of the feasible region are the intersection points of the lines.
Step 4: Calculate the profit at each corner point.
- Plug the coordinates of each corner point into the profit function P = 14x + 22y - 900 and calculate the profit.
Step 5: Determine the maximum profit.
- Compare the profits calculated in step 4 to find the corner point(s) that yield the maximum profit.
By following these steps, you can find the production levels that yield the maximum profit.
you were wrong
Apologies for any confusion caused. Let's go through the steps again to determine the production levels that yield the maximum profit.
Given:
Profit function: P = 14x + 22y - 900
Constraints:
- y - x ≤ 200
- x + 2y ≤ 1600
Step 1: Convert the given constraints into a standard form.
- y - x ≤ 200 is already in standard form.
- Rewrite x + 2y ≤ 1600 as -x - 2y ≥ -1600.
Step 2: Plot the feasible region by graphing the inequalities.
- Start by graphing the line y - x = 200.
- Find two points on this line by setting x = 0 and solving for y, and setting y = 0 and solving for x.
- Connect these points to draw the line.
- Choose a side of the line to represent the feasible region.
- Graph the line -x - 2y = -1600 in the same manner.
- The feasible region is the area bounded by these lines and the axes.
Step 3: Identify the corner points of the feasible region.
- The corner points of the feasible region are the intersection points of the lines.
Step 4: Calculate the profit at each corner point.
- Plug the coordinates of each corner point into the profit function P = 14x + 22y - 900 and calculate the profit.
Step 5: Determine the maximum profit.
- Compare the profits calculated in step 4 to find the corner point(s) that yield the maximum profit.
I apologize if my previous response was not clear or did not provide the correct answer. Please let me know if you need further clarification or assistance.