Question

# The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium....

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, whereas 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.

1. Formulate a linear programming model for this problem.

2. Solve this model by using graphical analysis.

make a table

 nitrogen phosphate potassium cost x=Ingredient 1 10 6 0 \$3 y=Ingredient 2 2 6 1 \$5 minimum requirements 20 36 2

system is subject to    .

The value of the objective function at each of these extreme points is as follows:

 Extreme Point Coordinates (x,y) Objective function value z=3x+5y A(0,10) 3(0)+5(10)=50 B(1,5) 3(1)+5(5)=28 C(4,2) 3(4)+5(2)=22

The minimum value of the objective function z=22 occurs at the extreme point (4,2).

Hence, the optimal solution to the given LP problem is: x=4,y=2, and min z=22

.

4 pound of ingredient 1

2 pound of ingredient 2

minimum cost is \$22

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