Question 1186248
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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.


<pre>
**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**.    <<<---===  <U>ANSWER</U>
</pre>

Solved.