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.
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?
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