The Kalo Fertilizer

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vocampo0811

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How to solve:

The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, while a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of fertilizer to meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost.

a. Formulate a linear programming model for this problem.

b. Solve this model by using graphical analysis.
 
Hello, and welcome to FMH! :)

Let's let \(x\) be the amount in pounds of ingredient 1, and \(y\) be the amount in pounds of ingredient 2 in our final mix.. In terms of \(x\) and \(y\), how many ounces of each of the 3 ingredients (nitrogen, phosphate and potassium) would be present in the final mix?
 
vocampo0811 said:
20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium
Click to expand...

That is the minimum requirement for those ingredients. Let's look at nitrogen. We are told each pound of ingredient 1 contributes 20 ounces of nitrogen, and each point of ingredient 2 contributes 2 ounces of nitrogen. If \(x\) and \(y\) represent the number of pounds of ingredient1 and ingredient 2 respectively in our final mix, then the number of ounces of nitrogen \(N\) in this mix will be:

[MATH]N=20x+2y\ge20[/MATH]
Can you now do the same for phosphate and potassium?
 
Each pound of ingredient 1 contributes to 10 ounces of nitrogen and 6 ounces of phosphate. Ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate and 1 ounce of potassium
So wouldn't it be N = 10x +2y ≥ 20,
Phosphate = 6x + 6y ≥ 26
Potassium = 1y ≥ 2
 
vocampo0811 said:
Each pound of ingredient 1 contributes to 10 ounces of nitrogen and 6 ounces of phosphate. Ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate and 1 ounce of potassium
So wouldn't it be N = 10x +2y ≥ 20,
Phosphate = 6x + 6y ≥ 26
Potassium = 1y ≥ 2
Click to expand...

You have a minor typo in your second inequality, but yes, we want:

[MATH]20x+2y\ge20[/MATH]
[MATH]6x+6y\ge36[/MATH]
[MATH]y\ge1[/MATH]
Or (after simplification):

[MATH]10x+y\ge10[/MATH]
[MATH]x+y\ge6[/MATH]
[MATH]y\ge1[/MATH]
Also, we should observe we cannot add negative amounts if either ingredient:

[MATH]0\le x[/MATH]
[MATH]0\le y[/MATH] (this is actually already covered above).

So, we will want to consider only quadrant I vertices of the intersection of the 3 conditions above...here is a plot this intersection, with vertices labeled...

fmh_0117.png

Can give the cost function in terms of \(x\) and \(y\)?
 
We are putting \(x\) pounds of ingredient 1, at a cost of 3 dollars/pound and \(y\) pounds of ingredients 2, at a cost of 5 dollars per pound into a bag of fertilizer. So what is the cost to us, in terms of \(x\) and \(y\), to do this?
 
Yes, so for which of the vertices in the graph is the cost function a minimum?
 
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