coffee blending

Question

The coffee peddler has 1440 lb of Sumatran coffee and 700 lb of Kona coffee. A batch of Hawaiian blend requires 8lb of Kona and 12 lbs of Sumatran and yields a profit of $90. A batch of classic blend requires 4 lbs and 16 lbs of Sumatran and yields $55 profit. how many batches of each kind should be made in order to maximize profit. what is the maximum profit?

 

(please show all work including charts and graphs)

Answer 1

Let's assume the peddler make H batches of Hawaiian blend and C batches of classic blend.

 

Then, the profit P = 90H + 55C -------(1)

 

If we want to find the maximum profit, we need to figure out the relationship between H and C.

 

There are other informations that are not used. The amounts of those two kinds of coffee are limited. In other words, the peddler can neither use more than 1440 lb of Sumatran coffee nor use more than 700 lb of Kona coffee.

 

Thus,

 

12H + 16C<=1440 -----------(2)

8H + 4C<=700       -----------(3)

 

When one of the two kinds of coffee is used up, the peddler gets the maximum profit because he couldn't make any kind of blend any more. 

 

Then, we need to consider two situations and compare the maximum profit in each situation to get the best solution for this problem.

 

1. The peddler uses up the 1440 lb of Sumatran coffee first.

 

    We change (2) into  12H + 16C=1440 

 

    C= (1440 - 12H) / 16  -------------(4)

    

    From (3) and (4), we get:

    8H + 4[(1440 - 12H) / 16]<=700

 

    => 5H+360<=700

    H<=68

 

    From (1) and (4), we get:

    P=90H + 55(1440 - 12H) / 16=48.75H + 4950  ------------(5)

 

    From (5), we can see that the more H he makes the more money he earns.

   

    In the first situation, the maximum profit happens when he make 68 batches of Hawaiian blend and 39        

    batches of classic blend.

 

    The profit is P1= $8265

 

2. The peddler uses up the 700 lb of Kona coffee first. 

    We apply similar method.

    

    Change (3) into  8H + 4C=700

    =>

    H=(700-4C)/8  ------------(6)

    

    Then

    P= 90[(700-4C)/8 ] + 55C=7875 + 10C

     

    From (2) and (6), we get:

    C<=39

    In the second situation, the maximum profit happens when he make 68 batches of Hawaiian blend and 39

    batches of classic blend.

    The profit is P2= $8265

 

Since P1=P2, the best solution is to make 68 batches of Hawaiian blend and 39 batches of classic blend.