Question 1186205: 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: 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 line. 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 3x + y = 150: Solving these gives x = 43, y = 19
* Intersection of 3x + y = 150 and x + y = 100: Solving these gives x = 25, y = 75
* Intersection of x + 7y = 175 and x + y = 100: Solving these gives x = 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.
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
c. Determine the best combination of the ingredients to minimize the 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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