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Spruce | Maple | |
---|---|---|
Planting Cost | $30 | $40 |
Area Required | 600 ft2 | 900 ft2 |
Carbon Dioxide Absorption | 650 lb/yr | 300 lb/yr |
Objective Function | |
---|---|
Verbal Expression | Algebraic Expression |
CO2 absorbed by Spruce trees (lb/yr) | 650x |
CO2 absorbed by Maple trees (lb/yr) | 300y |
Total CO2 absorbed by the trees (lb/yr) | 650x+300y |
Total CO2 absorbed by the trees is A (lb/yr) | A=650x+300y |
x=0
Zero Property of Multiplication
LHS/4=RHS/4
Constraint | Boundary Line | x=0 | y-intercept | y=0 | x-intercept |
---|---|---|---|---|---|
III | 3x+4y=210 | 3(0)+4y=210 | (0,52.5) | 3x+4(0)=210 | (70,0) |
IV | 2x+3y=150 | 2(0)+3y=150 | (0,50) | 2x+3(0)=150 | (75,0) |
Now we will plot the intercepts and connect them with line segments. The lines will be bound by the axes because the number of trees cannot be negative. Also, notice that the constraints are non-strict. Therefore, the lines will be solid.
The overlapping section of the graph above is the feasible region. Let's remove the unnecessary parts and indicate the vertices of the feasible region.
(I): LHS⋅2=RHS⋅2
(II): LHS⋅3=RHS⋅3
(II): Subtract (I)
(II): Distribute -1
(II): Subtract terms
(I): y=30
(I): Multiply
(I): LHS−240=RHS−240
(I): LHS/6=RHS/6
x=0, y=0
Zero Property of Multiplication
Add terms
Vertex | Objective Function | Absorption (lb/yr) |
---|---|---|
(0,0) | A=650(0)+300(0) | 0 |
(0,50) | A=650(0)+300(50) | 15000 |
(30,30) | A=650(30)+300(30) | 27500 |
(70,0) | A=650(70)+300(0) | 45500 |
As a result, 70 spruce trees should be planted to maximize the CO2 absorption.