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?
To solve this problem, we need to find the maximum value of the profit function P=14x+22y-900, while satisfying the production constraints.
First, let's rewrite the production constraint ×+2y<1600 as ×<1600-2y.
Next, we consider the constraint that the production of y can exceed × by no more than 200 units. This can be expressed as y<x+200.
To find the maximum profit, we need to find the values of x and y that maximize P while satisfying these two constraints.
Since P=14x+22y-900, we can rewrite it as P=-14(-x)-22(-y)+900, which gives us P=-14(-x)-22(-y)+900.
Now, we solve this problem using linear programming.
Step 1: Graph the two constraints on a coordinate plane.
The first constraint ×<1600-2y can be represented by a line with intercepts of (1600, 0) and (0, 800).
The second constraint y<x+200 can be represented by a line with intercepts of (0, 200) and (1400, 1600).
Step 2: Shade the region that satisfies both constraints.
The shaded region is the feasible region that satisfies both constraints.
Step 3: Identify the corner points of the shaded region.
The corner points of the shaded region are the points where the two lines intersect.
In this case, there are four corner points: (0, 200), (800, 400), (1200, 200), and (1200, 0).
Step 4: Evaluate the profit function at each corner point.
When we substitute the values of x and y into the profit function P=14x+22y-900, we get the following results:
- At (0, 200): P=14(0)+22(200)-900=4400.
- At (800, 400): P=14(800)+22(400)-900=15600.
- At (1200, 200): P=14(1200)+22(200)-900=17400.
- At (1200, 0): P=14(1200)+22(0)-900=13800.
Step 5: Compare the values of P at each corner point to find the maximum profit.
The maximum profit is achieved at (1200, 200) with a profit of $17,400.
Therefore, the production levels that yield the maximum profit are x=1200 units and y=200 units.
To find the production levels that yield maximum profit, we need to solve the given constraints mathematically.
Let's start by converting the inequality constraint into an equation:
x + 2y = 1600
Next, we need to consider the constraint that the production of y cannot exceed × by more than 200 units. We can express this constraint as:
y <= x + 200
Now, we will solve these two equations simultaneously to find the production levels that yield maximum profit.
1. Solve the first equation for x:
x = 1600 - 2y
2. Substitute the value of x from equation 1 into the second constraint:
y <= (1600 - 2y) + 200
3y <= 1800
y <= 600
So, the maximum value of y is 600.
3. Substitute the value of y into equation 1 to find the corresponding value of x:
x = 1600 - 2(600)
x = 1600 - 1200
x = 400
So, the maximum value of x is 400.
Therefore, the production levels that yield maximum profit are:
x = 400 units
y = 600 units
To find the production levels that yield the maximum profit, we need to optimize the profit function P=14x+22y-900 while considering the given constraints.
First, let's tackle the constraint related to the production of y not exceeding x by more than 200 units. We can write this constraint as y ≤ x + 200.
Next, let's address the constraint involving production levels: x + 2y < 1600.
Now, we can solve this problem using a mathematical technique called linear programming. The goal is to find the maximum value of the objective function (P) while satisfying all the constraints.
Step 1: Express the problem in terms of inequalities:
P = 14x + 22y - 900 (objective function)
Constraints:
1. y ≤ x + 200
2. x + 2y < 1600
Step 2: Graph the feasible region:
Graph the constraint equations on a coordinate plane and shade the region that satisfies all the constraints. The feasible region is the area where all constraints are simultaneously satisfied.
The feasible region would be a bounded region in the coordinate plane.
Step 3: Identify the vertices of the feasible region:
Find the coordinates of the vertices (corners) of the feasible region. These vertices are the possible solutions to the problem.
Step 4: Evaluate the objective function at each vertex:
Substitute the x and y values of each vertex into the objective function P = 14x + 22y - 900 and calculate the corresponding profit.
Step 5: Compare the profits obtained at each vertex:
Identify the vertex that yields the maximum profit. The production levels corresponding to that vertex will give you the optimal solution.
By following these steps, you should be able to find the production levels that yield the maximum profit in this scenario.