A factory can produce two products x and y with a profit approximated by P=14×+22y-900. The production of y can exceed × by no more than 200 units. Moreover, production levels are limited by the formula ×+2y<1600. What production levels yield maximum profit?
×=400; y=600
×=0; y=0
×=1,600; y=0
×=0; y=200
The maximum profit can be achieved by finding the vertex of the feasible region defined by the production limits.
First, let's consider the production limits:
1) Production of y can exceed × by no more than 200 units.
This constraint can be represented as y - x ≤ 200.
2) Production levels are limited by the formula ×+2y<1600.
This constraint can be represented as x + 2y ≤ 1600.
Now, let's plot these constraints on a graph:
To plot the constraint y - x ≤ 200, we can rewrite it as x = y - 200.
To plot the constraint x + 2y ≤ 1600, we can rewrite it as y = (1600 - x)/2.
Here is the graph:
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To find the maximum profit, we need to find the vertex of the feasible region. This can be done by evaluating the profit function P at the vertices of the feasible region.
The vertices of the feasible region are:
A: x = 0, y = 0
B: x = 0, y = 200
C: x = 400, y = 600
D: x = 1200, y = 200
Evaluating the profit function P at each vertex gives us the following profits:
A: P = 14(0) + 22(0) - 900 = -900
B: P = 14(0) + 22(200) - 900 = 1300
C: P = 14(400) + 22(600) - 900 = 22100
D: P = 14(1200) + 22(200) - 900 = 26600
Therefore, the production levels that yield maximum profit are:
x = 1200 and y = 200.
To find the production levels that yield maximum profit, we need to optimize the profit function P=14x+22y-900 subject to the given constraints.
First, let's consider the constraint that the production of y can exceed x by no more than 200 units: y - x ≤ 200
Next, let's consider the constraint for production levels: x + 2y < 1600.
To solve this problem, we can use the method of linear programming.
Step 1: Convert the inequality constraint x + 2y < 1600 into an equation:
x + 2y = 1600
Step 2: Graph the system of inequalities and find the region of feasible solutions.
Step 3: Calculate the corner points of the feasible region, which are the intersection points of the lines representing the constraints.
Solving the system of equations:
x + 2y = 1600
y - x = 200
By solving these equations, we get two corner points:
(400, 600) and (0, 200)
Step 4: Substitute the corner points into the profit function to evaluate the profit at each corner point:
For (400, 600):
P = 14(400) + 22(600) - 900
P = 5600 + 13200 - 900
P = 18900
For (0, 200):
P = 14(0) + 22(200) - 900
P = 0 + 4400 - 900
P = 3500
Step 5: Compare the profits at each corner point to identify the production levels that yield the maximum profit.
From the calculations, we can see that the production levels of x = 400 and y = 600 yield a maximum profit of P = 18900.
Therefore, the correct answer is: × = 400; y = 600.
To find the production levels that yield maximum profit, we need to consider the given constraints.
The first constraint states that the production of y cannot exceed x by more than 200 units. This can be written mathematically as y <= x + 200.
The second constraint states that the sum of x and 2y must be less than 1600. This can be written mathematically as x + 2y < 1600.
To solve this problem, we can apply the method of linear programming. We can graph the feasible region defined by these inequalities, and then evaluate the profit function within this region to find the maximum profit.
Let's start by graphing the feasible region:
- Plot the equation y = x + 200 as a line.
- Plot the equation x + 2y = 1600 as a line.
After graphing the lines, shade the region that satisfies both constraints.
Next, we need to evaluate the profit function P = 14x + 22y - 900 at the corner points of the shaded region. These corner points are the vertices of the feasible region.
Let's evaluate the profit function at each corner point:
1. (x=400, y=600): P = 14(400) + 22(600) - 900 = 8,400
2. (x=0, y=0): P = 14(0) + 22(0) - 900 = -900
3. (x=1600, y=0): P = 14(1600) + 22(0) - 900 = 19,700
4. (x=0, y=200): P = 14(0) + 22(200) - 900 = 2,300
Comparing the profits at each corner point, we can see that the maximum profit is achieved at (x=400, y=600) with a profit of 8,400. Therefore, the production levels that yield maximum profit are x = 400 and y = 600.
So, the answer is: x = 400, y = 600.