Question 1186248: The Intellectual Company produces a chemical solution used for cleaning carpets. This chemical is made from a mixture of two other chemicals which contain cleaning agent X and cleaning agent Y. Their product must contain 175 units of agent X and 150 units of agent Y and weigh at least 100 pounds. Chemical A costs ₱ 8 per pound, while chemical B costs ₱ 6 per pound. Chemical A contains one unit of agent X and three units of agent Y. Chemical B contains seven units of agent X and one unit of agent Y.
a. Set up the following:
i. Variables
ii. Constraints
iii. Objective Function
b. Find the minimum cost
c. Determine the best combination of the ingredients to minimize the cost.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to set up and solve this linear programming problem:
**a. Set up:**
**i. Variables:**
* x = pounds of chemical A
* y = pounds of chemical B
**ii. Constraints:**
* Agent X: 1x + 7y >= 175
* Agent Y: 3x + 1y >= 150
* Weight: x + y >= 100
* Non-negativity: x >= 0, y >= 0 (We can't have negative amounts of chemicals.)
**iii. Objective Function:**
We want to minimize the cost, which is given by:
Cost (C) = 8x + 6y
**b. Find the minimum cost:**
To find the minimum cost, we need to graph the constraints and find the feasible region. The optimal solution will occur at one of the vertices (corners) of the feasible region.
1. **Graph the constraints:** Treat each inequality as an equation and plot the lines. Then shade the appropriate region based on the inequality.
* x + 7y = 175
* 3x + y = 150
* x + y = 100
2. **Identify the feasible region:** The feasible region is the area where all the shaded regions overlap.
3. **Find the vertices:** The vertices of the feasible region are the points where the constraint lines intersect. You'll need to solve systems of equations to find these intersection points. The relevant vertices are usually where two constraints intersect.
* Intersection of x + 7y = 175 and 3x + y = 150: Solving these equations gives x = 43, y = 19
* Intersection of 3x + y = 150 and x + y = 100: Solving these equations gives x = 25, y = 75
* Intersection of x + 7y = 175 and x + y = 100: Solving these equations gives x = 75/6 = 12.5, y = 87.5
4. **Evaluate the objective function at each vertex:**
* C(43, 19) = 8(43) + 6(19) = 344 + 114 = 458
* C(25, 75) = 8(25) + 6(75) = 200 + 450 = 650
* C(12.5, 87.5) = 8(12.5) + 6(87.5) = 100 + 525 = 625
5. **Determine the minimum cost:** The minimum cost is the smallest value of the objective function at the vertices.
The minimum cost is ₱458.
**c. Determine the best combination of ingredients:**
The best combination of ingredients is the (x, y) values that correspond to the minimum cost.
The minimum cost of ₱458 occurs when x = 43 and y = 19.
Therefore, the best combination is 43 pounds of chemical A and 19 pounds of chemical B.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
The Intellectual Company produces a chemical solution used for cleaning carpets.
This chemical is made from a mixture of two other chemicals which contain cleaning agent X and cleaning agent Y.
Their product must contain 175 units of agent X and 150 units of agent Y and weigh at least 100 pounds.
Chemical A costs ₱ 8 per pound, while chemical B costs ₱ 6 per pound.
Chemical A contains one unit of agent X and three units of agent Y.
Chemical B contains seven units of agent X and one unit of agent Y.
a. Set up the following:
i. Variables
ii. Constraints
iii. Objective Function
b. Find the minimum cost
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The solution in the post by @CPhill, giving the answer
x = 43 pounds of chemical A and y = 19 pounds of chemical B
is INCORRECT. It can be easily disproved, since x + y = 43 + 19 = 62, which is less than 100.
Thus the restriction x+y >= 100 of the problem is not satisfied: it is FAILED, instead.
The cause is that @CPhill incorrectly determined the feasibility domain and used WRONG vertices for estimations.
Below is my solution, proper and correct.
**a. Set up:**
**i. Variables:**
* x = pounds of chemical A
* y = pounds of chemical B
**ii. Constraints:**
* Agent X: x + 7y ≥ 175
* Agent Y: 3x + y ≥ 150
* Weight: x + y ≥ 100
* Non-negativity: x ≥ 0, y ≥ 0
**iii. Objective Function:**
Minimize Cost (C) = 8x + 6y
**b. Find the minimum cost:**
1. **Graph the constraints:** Treat each inequality as an equation and plot the lines. Shade the appropriate region based on the inequality.
For example, for x + 7y ≥ 175, plot the line x + 7y = 175, and shade the region *above* and to the *right* of the lines.
Do this for all constraints.
2. **Identify the feasible region:** The feasible region is the area where all the shaded regions overlap.
3. **Find the vertices:** The vertices of the feasible region are the points where the constraint lines intersect.
Solve systems of equations to find these intersection points. The relevant vertices are:
* Intersection of x + 7y = 175 and y = 0: x = 175, y = 0
* Intersection of x + 7y = 175 and x + y = 100: Solving these gives x = 87.5, y = 12.5
* Intersection of 3x + y = 150 and x + y = 100: Solving these gives x = 25, y = 75
* Intersection of 3x + y = 175 and x = 0: Solving these gives x = 0, y = 175
4. **Evaluate the objective function at each vertex:**
* C(175, 0) = 8*175 + 6*0 = 1400
* C( 87.5, 12.5) = 8*81.5 + 6*12.5 = 727
* C(25, 75) = 8*25 + 6*75 = 650
* C(0, 175) = 8*0 + 6*175 = 1050
5. **Determine the minimum cost:** The minimum cost is the smallest value of the objective function.
The minimum cost is ₱650.
**c. Determine the best combination of ingredients:**
The best combination of ingredients is the (x, y) values that correspond to the minimum cost.
The minimum cost of ₱650 occurs when x = 25 and y = 75.
Therefore, the best combination is **25 pounds of chemical A and 75 pounds of chemical B**. <<<---=== ANSWER
Solved.
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