Question

Resource allocation. A coffee manufacturer uses Colombian and Brazilian coffee beans to produce two? blends, robust and mild. A pound of the robust blend requires 12 ounces of Colombian beans and 4 ounces of Brazilian beans. A pound of the mild blend requires 6 ounces of Colombian beans and 10 ounces of Brazilian beans. Coffee is shipped in 138pound burlap bags. The company has 65 bags of Colombian beans and 30 bags of Brazilian beans on hand. How many pounds of each blend should they produce in order to use all the available? beans?

Robustequals=	nothing lbs
Mildequals=	nothing lbs

Answer 1

Let the coffee manufacturer produce x lbs of robust blend and y lbs of mild blend. The company has 65*138 = 8970 lbs of Colombian beans and 30*138 = 4140 lbs of Brazilian beans on hand.

Since 1 lb of the robust blend requires 12 ounces (i.e. 12/16 = ¾ lb) of Colombian beans and 4 ounces (i.e. 4/16 = ¼ lb) of Brazilian beans and since 1 lb of the mild blend requires 6 ounces (i.3. 6/16 = 3/8 lb) of Colombian beans and 10 ounces ( i.e. 10/16 = 5/8 lb) of Brazilian beans, hence (3/4)x + (3/8)y = 8970 or, on multiplying both the sides by 8/3, we get 2x+ y = 23920…(1)

Also, (1/4)x + (5/8)y = 4140 or, on multiplying both the sides by 8, we get 2x+5y = 33120…(2).

The augmented matrix of the above linear system is A =

2	1	23920
2	5	33120

To solve the above linear system, we will reduce A to its RREF as under:

Multiply the 1st row by ½

Add -2 times the 1st row to the 2nd row

Multiply the 2nd row by ¼

Add -1/2 times the 2nd row to the 1st row

Then the RREF of A is

1	0	10810
0	1	2300

Hence x = 10810 and y = 2300.

Thus, the quantities of Robust and the Mild blend to be produced by the coffee manufacturer, in order to use all the available? beans, are as under:

Robust Blend	10810 lbs
Mild Blend	2300 lbs