Pearson Algebra 2 Common Core, 2011
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Pearson Algebra 2 Common Core, 2011 View details
4. Linear Programming
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Exercise 13 Page 160

a We are given a table that represents the Spruce and Maple Tree Data.
Spruce Maple
Planting Cost
Area Required
Carbon Dioxide Absorption lb/yr lb/yr
Let be the number of spruces and be the number of maples. We can write the first two constraints based on the fact that the number of trees cannot be negative.
Since the city has to spend on planting, the total cost of the trees will be less than or equal to the city's budget.
The available area for planting in the city is Therefore, the total area covered by the trees will be less than or equal to city's available area.
b The quantity we are trying to maximize or minimize is modeled with an objective function. In this case, we are trying to maximize the carbon dioxide absorption. Therefore, the objective function can be written using the third row of the table.
Objective Function
Verbal Expression Algebraic Expression
absorbed by Spruce trees (lb/yr)
absorbed by Maple trees (lb/yr)
Total absorbed by the trees (lb/yr)
Total absorbed by the trees is
c Let's start with the first two constraints. We will first determine their boundary lines by replacing the inequality sign with the equals sign.
The boundary lines of the first two constraints are the axes. Since both and are greater than or equal to we will shade Quadrant I. Notice that the inequalities are non-strict, so the boundary lines will be solid.
Next, we will graph Constraint III and IV by finding their boundary lines.
To graph their boundary lines, we will find the and intercepts of the lines. We will first substitute for into Boundary Line III and solve it for to find its intercept.
The intercept of Boundary Line III is the point Proceeding in the same way, we can find the intercepts of both lines.
Constraint Boundary Line intercept intercept
III
IV

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.

Next, we will test the point to determine which region we should shade. Let's start with Constraint III.
For Constraint III, we will shade the region that contains the point.
Now we will test the point for Constraint IV.
Therefore, we will again shade the region that contains the test point.

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.

The coordinates of three of the four vertices are determined. The fourth vertex is the point of intersection of Constraint III and IV. To determine it we will first write a system of equations.
To solve the system, we will equate the coefficients of and use the Elimination Method.
The coordinate of the point is By substituting for into the equation, we can find the coordinate.
Thus the fourth vertex is
d To find the number of each tree for the maximum absorption, we will substitute the vertices into the objective function. Thus, we can determine the maximum value of Let's begin with the vertex
For the vertex the absorption will be lb/yr. We can find the absorption for the other vertices proceeding in the same way.
Vertex Objective Function Absorption (lb/yr)

As a result, spruce trees should be planted to maximize the absorption.