Algebra 2 Linear Programming Help Please

Question

Birmingham Coffee sells 2 blends of beans: Bama Blend and Auburn Blend. Bama Blend is 1/4 Costa Rican beans and 1/3 Colombian beans. Auburn Blend is 1/2 Costa Rican beans and 1/2 Columbian beans. Profit on the Bama Blend is $4 per pound, while profit on the Auburn Blend is $3 per pound. If Birmingham Coffee receives 300 pounds of Costa Rican beans and 400 pounds of Columbian beans daily, how many pounds of each blend should Birmingham Coffee prepare each day is it wants to maximize the profit? What will the maximum profit be?

A) Choose a variable for each product.
Let a be the number of pounds of Auburn Blend and b be the .

B) Form a total profit statement.
The problem gives us the profit for each blend, so the total profit is
P(a, b) = $ a + $ b

C) Write your constraints as a system of inequalities.
The fewest pounds that can be produced is . Therefore,
a ≥ and b ≥

The maximum number of pounds for each type of bean is represented by
a + b ≤ (Costa Rican beans)
a b ≤ (Columbian beans)

Answer 1

For part B... to find the profit you multiply the profit per pound times the number of pounds sold.

 

You make $4 per pound of Bama and $3 per pound of Auburn.

 

So the total profit will be $4 times the pounds of Bama sold  plus $3 times the pounds of Auburn sold.

 

P = 4B + 3A

 

That is what we want to maximize.

 

 

Part C:

Now we set up the inequalities.

They tell us the following:

Bama is 1/4 Costa Rican and 1/3 Colombian

Auburn is 1/2 Costa Rican and 1/2 Colombian

If we add the amount of Costa Rican beans from both blends we cannot get more than 300 lbs.
So ¼(B) plus ½(A) ≤ 300.

If we add the amount of Colombian beans from both blends we cannot get more than 400 lbs.
So 1/3(B) + ½(A) ≤ 400.

Then we have 2 more:
The amount of Bama cannot be negative so B ≥ 0.
The amount of Auburn cannot be negative so A ≥ 0.

 

Now you will be ready to graph those inequalities and answer the rest of the question.

 

If you have trouble graphing those let me know and I'll make a video and post it for you.  :-)

 

Also if you have questions about any of it to this point, let me know and I'll try to explain better.

 

 

 

 

 

Bama           1/4 Costa Rican             1/3 Colombian

 

 

Auburn         1/2 Costa Rican             1/2 Colombian

 

                    

Total available  300 lb                          400 lb

 

 

 

for our blends