word problem

Question

It's a linear programming word problem.

 

For a given week, Greg's Coffee House has available 1728 ounces of A grade coffee and 1760 ounces of B grade coffee. These are blended into 1-pound packages as follows: an economy blend that contains 2 ounces of A grade coffee and 10 ounces of B grade coffee, and a superior blend that contains 9 ounces of A grade coffee and 2 ounces of B grade coffee. (The remainder of each blend is made of filler ingredients.) There is a $3 profit on each economy blend package sold and a $4 profit on each superior blend package sold. Assuming that the store is able to sell as many blends as it makes, how many packages of each blend should it make to maximize its profit for the week?

Answer 1

For this question you need to start off with a chart.

 

Economy (x)     Superior (y)             Maximum # of Ounces

 

A Grade                 2                      9                                    1728

B Grade                 10                    2                                    1760

 

Using this chart and other constraints (like not being able to have negative ounces of coffee), turn these numbers for each grade into inequalities and graph.

 

A Grade: 2x+9y≤1728

B Grade: 10x+2y≤1760

x<0

y<0

 

Graph all four of these inequalities and find where all the shaded areas overlap. Should be a small four shaded region nearest to (0,0) Next, find all the points of intersection around the shaded region (where any 2 lines meet). You should have four points as follows:

 

(0,192)

(144,160)

(176,0)

(0,0)

 

Plug these values into your profit equation to find out which combination makes the most profit: P=3x+4y

P=3(0)+4(192)

P=768

 

P=3(144)+4(160)

P=1072

 

P=3(176)+4(0)

P=528

 

P=3(0)+4(0)

P=0

 

The most profit comes when 144 economy blends and 160 superior blends are made.