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? Choose from answers.
- ×=400; y=600
- ×=0; y=0
- ×=1,600; y=0
- ×=0; y=200
To find the production levels that yield the maximum profit, we can use optimization techniques.
First, let's set up the constraints based on the given information. We have two constraints:
1) The production of y can exceed x by no more than 200 units: y - x ≤ 200
2) Production levels are limited by the formula x + 2y < 1600
Now let's analyze the options:
- Option 1: x = 400; y = 600
- Option 2: x = 0; y = 0
- Option 3: x = 1,600; y = 0
- Option 4: x = 0; y = 200
We need to check if these options satisfy the constraints.
- Option 1: (400) + 2(600) = 1600, which satisfies the second constraint. However, y - x = 200, which violates the first constraint.
- Option 2: (0) + 2(0) = 0, which satisfies the second constraint. But y - x = 0 - 0 = 0, which violates the first constraint.
- Option 3: (1600) + 2(0) = 1600, which violates the second constraint.
- Option 4: (0) + 2(200) = 400, which satisfies the second constraint. y - x = 200 - 0 = 200, which satisfies the first constraint.
Only option 4, x = 0; y = 200, satisfies both constraints.
Therefore, the production levels that yield the maximum profit are x = 0 and y = 200.
To find the production levels that yield maximum profit, we need to find the maximum values of x and y that satisfy the given conditions.
From the first condition, we have y ≤ x + 200.
Substituting this into the second condition, we get x + 2(x + 200) < 1600.
Simplifying, we get x + 2x + 400 < 1600.
Combining like terms, we get 3x + 400 < 1600.
Subtracting 400 from both sides, we get 3x < 1200.
Dividing by 3, we get x < 400.
So, the maximum value for x is 400.
Substituting this into the first condition, we get y ≤ 400 + 200.
Simplifying, we get y ≤ 600.
So, the maximum value for y is 600.
Therefore, the production levels that yield maximum profit are x = 400 and y = 600.
The answer is: x = 400; y = 600.
To find the production levels that yield maximum profit, we need to optimize the profit function P = 14x + 22y - 900, subject to the constraints:
1) Production of y can exceed x by no more than 200 units: y - x ≤ 200.
2) Production levels are limited by the formula x + 2y < 1600.
To solve this problem, we will use the method of linear programming.
Step 1: Convert the inequality constraints into equalities:
y - x = 200 (Constraint 1)
x + 2y = 1600 (Constraint 2)
Step 2: Solve the system of equations formed by the equalities:
From Constraint 2, we can express x in terms of y:
x = 1600 - 2y
Substituting this expression for x into Constraint 1:
y - (1600 - 2y) = 200
3y = 1800
y = 600
Substituting the value of y back into x = 1600 - 2y:
x = 1600 - 2(600)
x = 400
Therefore, the production levels that yield maximum profit are:
x = 400 and y = 600.
Therefore, the correct answer is:
- x = 400; y = 600.