Math  /  Algebra

Question1. Khorshid factory produces two producers) A B) and the working hours in this factory are equivalent to 8 hours divided into producers and raw materials available 18 tons, if you know that product AA needs one hour of working hours and 2 kilos of raw materials while product BB needs one hour of work and 3 kilos of raw materials and the profit return from product A$2A \$ 2 while the profit return from product B$3B \$ 3.
Required
1. Define the profit function.
2. Define limitations.
3. Draw the solution area.
4. Select Solution points.
5. Find the productive mix that maximizes profits from the two commodities.
6. Determine the type of relationship between the two products.

Studdy Solution

STEP 1

1. Khorshid factory produces two products: A and B.
2. The factory operates for 8 hours.
3. There are 18 tons (18,000 kilos) of raw materials available.
4. Product A requires 1 hour of work and 2 kilos of raw materials.
5. Product B requires 1 hour of work and 3 kilos of raw materials.
6. The profit from product A is 2.<br/>7.TheprofitfromproductBis2.<br />7. The profit from product B is 3.
8. We need to define the profit function, limitations, draw the solution area, select solution points, find the productive mix that maximizes profits, and determine the type of relationship between the two products.

STEP 2

1. Define the profit function.
2. Define the limitations (constraints).
3. Draw the solution area (feasible region).
4. Select solution points.
5. Find the productive mix that maximizes profits.
6. Determine the type of relationship between the two products.

STEP 3

Define the profit function.
Let x x be the number of product A produced, and y y be the number of product B produced. The profit function P P can be defined as:
P=2x+3y P = 2x + 3y

STEP 4

Define the limitations (constraints).
1. Time constraint: x+y8 x + y \leq 8 (since each product requires 1 hour of work)
2. Raw material constraint: 2x+3y18000 2x + 3y \leq 18000 (since product A requires 2 kilos and product B requires 3 kilos of raw materials)
3. Non-negativity constraints: x0 x \geq 0 and y0 y \geq 0

STEP 5

Draw the solution area (feasible region).
Plot the constraints on a graph with x x on the horizontal axis and y y on the vertical axis. The feasible region is the area where all constraints overlap.

STEP 6

Select solution points.
Identify the vertices (corner points) of the feasible region. These points are potential candidates for maximizing the profit function.

STEP 7

Find the productive mix that maximizes profits.
Evaluate the profit function P=2x+3y P = 2x + 3y at each vertex of the feasible region. The vertex that gives the highest value of P P is the optimal solution.

STEP 8

Determine the type of relationship between the two products.
Analyze the constraints and the profit function to determine if the products are complementary, substitutes, or independent.

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