Problems A farmer uses two types of fertilizers. A 75-lb. bag of Fertilizer A contains 10 lbs. of potassium, 2 lbs. of phosphorus, and 6 lbs. of nitrogen. A 75-lb. bag of Fertilizer B contains 5 lbs. each of potassium, phosphorus, and nitrogen. The minimum requirements for a field are 420 lbs. of potassium, 240 lbs. of phosphorus, and 360 lbs. of nitrogen. If a 75-lb. bag of Fertilizer A costs $35 and a 75-lb. bag of Fertilizer B costs $25, find the amount of each fertilizer the farmer should use to minimize her cost while still meeting the minimum requirements. What is the cost? What are the extra resources? Part 1 of 3 Part 1: Set up the linear programming problem. (a.) Define the variables. {x = 1 pounds of Fertilizer A used (x = number of Fertilizer A bags used = Fertilizer A {x = Fertilizer B (x = cost of Fertilizer A (in dollars) {x = cost of Fertilizer B (in dollars) (b.) State the objective. The objective is to minimize ✔ minimize cost✔ cost. (c.) Write the objective function. C = 35x+25y (d.) Write the constraints. 35x + 25y 10x+5y420 2x+5y240 42010x+5.. (potassium requirement) Y 2402x+5y (phosphorus requirement) 6x+5y360 3606 +5. y (nitrogen requirement) x≥ 0, y≥ 0 Part 2 of 3 Part 2: Solve using the appropriate simplex method. Show work by switching to a maximum LP problem, creating the initial simplex tableau, circling the first pivot element, and giving the final simplex tableau. (e.) Switch to a maximization LP problem. Maximize P = -C = -35x-25y (f.) Set up the initial simplex tableau. x y u V W P Constants -10 -5 1 0 0 -420 -2 -5 0 0 0 -240 -6 -5 0 1 0 -360 35 25 ✔ 0 0 0 1 0 (g.) Give the location of the first pivot. Row: Row 1 Column: Column 1 (h.) Give the final simplex tableau. < x y u v W P Constants 1 0 -1 0 -1 × 0 42 0 |1 -1 0 -1 × 0 0 ☑ 0 0 1 1 -1 × 0 10 ☑ 0 10 -1 0 -1 × 1 0
Problems A farmer uses two types of fertilizers. A 75-lb. bag of Fertilizer A contains 10 lbs. of potassium, 2 lbs. of phosphorus, and 6 lbs. of nitrogen. A 75-lb. bag of Fertilizer B contains 5 lbs. each of potassium, phosphorus, and nitrogen. The minimum requirements for a field are 420 lbs. of potassium, 240 lbs. of phosphorus, and 360 lbs. of nitrogen. If a 75-lb. bag of Fertilizer A costs $35 and a 75-lb. bag of Fertilizer B costs $25, find the amount of each fertilizer the farmer should use to minimize her cost while still meeting the minimum requirements. What is the cost? What are the extra resources? Part 1 of 3 Part 1: Set up the linear programming problem. (a.) Define the variables. {x = 1 pounds of Fertilizer A used (x = number of Fertilizer A bags used = Fertilizer A {x = Fertilizer B (x = cost of Fertilizer A (in dollars) {x = cost of Fertilizer B (in dollars) (b.) State the objective. The objective is to minimize ✔ minimize cost✔ cost. (c.) Write the objective function. C = 35x+25y (d.) Write the constraints. 35x + 25y 10x+5y420 2x+5y240 42010x+5.. (potassium requirement) Y 2402x+5y (phosphorus requirement) 6x+5y360 3606 +5. y (nitrogen requirement) x≥ 0, y≥ 0 Part 2 of 3 Part 2: Solve using the appropriate simplex method. Show work by switching to a maximum LP problem, creating the initial simplex tableau, circling the first pivot element, and giving the final simplex tableau. (e.) Switch to a maximization LP problem. Maximize P = -C = -35x-25y (f.) Set up the initial simplex tableau. x y u V W P Constants -10 -5 1 0 0 -420 -2 -5 0 0 0 -240 -6 -5 0 1 0 -360 35 25 ✔ 0 0 0 1 0 (g.) Give the location of the first pivot. Row: Row 1 Column: Column 1 (h.) Give the final simplex tableau. < x y u v W P Constants 1 0 -1 0 -1 × 0 42 0 |1 -1 0 -1 × 0 0 ☑ 0 0 1 1 -1 × 0 10 ☑ 0 10 -1 0 -1 × 1 0
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter6: Systems Of Linear Equations And Inequalities
Section6.1: Graphing Systems Of Equations
Problem 59PFA
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Transcribed Image Text:Problems
A farmer uses two types of fertilizers. A 75-lb. bag of Fertilizer A contains 10 lbs. of potassium, 2 lbs. of phosphorus, and 6 lbs. of nitrogen. A 75-lb. bag of Fertilizer B contains 5 lbs. each of potassium, phosphorus, and nitrogen. The minimum requirements for a field are 420 lbs. of potassium, 240 lbs. of phosphorus, and 360 lbs. of nitrogen. If a 75-lb. bag of Fertilizer A costs $35 and a 75-lb. bag of
Fertilizer B costs $25, find the amount of each fertilizer the farmer should use to minimize her cost while still meeting the minimum requirements. What is the cost? What are the extra resources?
Part 1 of 3
Part 1: Set up the linear programming problem.
(a.) Define the variables.
{x =
1
pounds of Fertilizer A used
(x = number of Fertilizer A bags used
= Fertilizer A
{x = Fertilizer B
(x = cost of Fertilizer A (in dollars)
{x = cost of Fertilizer B (in dollars)
(b.) State the objective.
The objective is to minimize ✔
minimize
cost✔
cost.
(c.) Write the objective function.
C = 35x+25y
(d.) Write the constraints.
35x + 25y
10x+5y420
2x+5y240
42010x+5..
(potassium requirement)
Y
2402x+5y
(phosphorus requirement)
6x+5y360
3606 +5. y
(nitrogen requirement)
x≥ 0, y≥ 0
Part 2 of 3
Part 2: Solve using the appropriate simplex method. Show work by switching to a maximum LP problem, creating the initial simplex tableau, circling the first pivot element, and giving the final simplex tableau.
(e.) Switch to a maximization LP problem.
Maximize P = -C =
-35x-25y
(f.) Set up the initial simplex tableau.
x
y
u
V
W
P
Constants
-10
-5
1
0
0
-420
-2
-5
0
0
0
-240
-6
-5
0
1
0
-360
35
25 ✔
0
0
0
1
0
(g.) Give the location of the first pivot.
Row: Row 1
Column: Column 1
(h.) Give the final simplex tableau.
<
x
y
u
v
W
P
Constants
1
0
-1
0
-1 × 0
42
0
|1
-1
0
-1
× 0
0
☑
0
0
1
1
-1
× 0
10
☑
0
10
-1
0
-1 × 1
0
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