Axia College Material

Appendix D

Landscape Design

Landscape designers often use coordinate geometry and algebra as they help their clients. In many regions, landscape design is a growing field. With the increasing popularity of do-it-yourself television shows, however, many homeowners are becoming amateur landscape artists.

Suppose you are a homeowner getting ready to sell your home. You realize that there are some landscaping problems that you want to address so that your home will sell quickly and you can get the best price. Since deciding to landscape your backyard, you have realized there are many things to consider, such as budget, time, and space.

Application Practice

Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting.

1. You are planning to spend no less than $6,000 and no more than $10,000 on your landscaping project.
a) Write an inequality that demonstrates how much money you will be willing to spend on the project.

b) Suppose you want to cover the backyard with decorative rock and plant some trees as the first phase of the project. You need 30 tons of rock to cover the area. If each ton cost $60 and each tree is $84, what is the maximum number of trees you can buy with a budget for rock and trees of $2,500? Write an inequality that illustrates the problem and solve. Express your answer as an inequality and explain how you arrived at your answer.



c) Would 5 trees be a solution to the inequality in part b? Justify your answer.

2. The coordinate graph of the backyard shows the location of trees, plants, the patio, and utility lines. (If necessary, you may copy and paste the image to another document and enlarge it.)
a) What are the coordinates of Tree A? Plant B? Plant C? Patio D? Plant E? Plant F?

b) The water line is given by the equation
.
Suppose you want to put a pink flamingo lawn ornament in your backyard, but you want to avoid placing it directly over the water line, in case you need to excavate the line for repairs in the future. Could you place it at the point (-4,-10)?

c) What is the slope and y-intercept of the line in part b? How do you know?

d) Suppose you want to add a sprinkler system, and the location of one section of the sprinkler line can be described by the equation

.

Complete the table for this equation.

x y (x,y)
-1
-2
-4
2
8

e) What objects might be in the way as you lay the pipe for the sprinkler?

1.

a) The inequality that demonstrates how much money you will be willing to spend on the project is:

$6,000 ≤ (amount you are willing to spend) ≤ $10,000

b) Let's define:
TR = number of trees
CR = cost of each ton of rock = $60
CT = cost of each tree = $84

The maximum number of trees you can buy with a budget for rock and trees of $2,500 can be determined by solving the following inequality:

30 * CR + TR * CT ≤ $2,500

Substituting the known values:

30 * 60 + TR * 84 ≤ $2,500

1800 + 84TR ≤ $2,500

84TR ≤ $2,500 - $1800

84TR ≤ $700

TR ≤ $700 / 84

TR ≤ 8.33

Since the number of trees must be a whole number, the maximum number of trees you can buy is 8.

c) To determine if 5 trees would be a solution to the inequality in part b, substitute TR = 5 into the inequality equation:

30 * 60 + 5 * 84 ≤ $2,500

1800 + 420 ≤ $2,500

2,220 ≤ $2,500

Since the inequality holds true, 5 trees would be a solution to the inequality in part b.

2.
a) The coordinates of the given points are as follows:
Tree A: (3, 4)
Plant B: (-2, 6)
Plant C: (1, -2)
Patio D: (-5, -1)
Plant E: (0, -5)
Plant F: (2, 2)

b) To determine if the point (-4, -10) is directly over the water line, substitute the coordinates (-4, -10) into the equation of the water line:

-10 = -2(-4) + 6

-10 = 8 + 6

-10 = 14

Since -10 does not equal 14, you can place the pink flamingo lawn ornament at the point (-4, -10).

c) The equation of the water line is given as:

y = -2x + 6

The slope of the line is -2, which can be determined from the coefficient of x in the equation. The y-intercept is 6, which can be determined from the constant term in the equation. This can be verified by comparing the equation to the general form y = mx + b, where m is the slope and b is the y-intercept.

d) Complete the table for the equation:

x | y | (x,y)
-1 | 8 | (-1, 8)
-2 | 4 | (-2, 4)
-4 | -2 | (-4, -2)
2 | -2 | (2, -2)
8 | 14 | (8, 14)

e) The objects that might be in the way as you lay the pipe for the sprinkler are not specified in the given information, so it is not possible to identify them without further details.

1.

a) The inequality that demonstrates how much money you will be willing to spend on the project is:
$6,000 ≤ x ≤ $10,000, where x represents the amount of money you will spend on the landscaping project.

b) To find the maximum number of trees you can buy with a budget of $2,500, you need to consider the cost of the decorative rock and the trees. Let's assume the number of trees you can buy is represented by t.

The cost of the decorative rock is $60 per ton, and you need 30 tons of rock, so the cost of the rock is 30 * $60 = $1800.
The cost of each tree is $84.
Therefore, the inequality that illustrates the problem is: 84t + 1800 ≤ 2500.

To solve the inequality, subtract 1800 from both sides:
84t ≤ 2500 - 1800
84t ≤ 700

Divide both sides of the inequality by 84:
t ≤ 700 / 84
t ≤ 8.3333 (approximately)

Since you cannot have a fraction of a tree, the maximum number of trees you can buy with a budget for rock and trees of $2,500 is 8.

c) To determine if 5 trees would be a solution to the inequality in part b, substitute 5 for t in the inequality:
84(5) + 1800 ≤ 2500
420 + 1800 ≤ 2500
2220 ≤ 2500

Since 2220 is less than or equal to 2500, 5 trees would be a solution to the inequality.

2.
a) The coordinates of Tree A, Plant B, Plant C, Patio D, Plant E, and Plant F are not given in the provided information. You may need to refer to a separate document or image to find the specific coordinates.

b) The equation of the water line is not provided in the given information.

c) Without the equation for the water line, we cannot determine if (-4, -10) is directly over the water line.

d) The table for the equation of the sprinkler line is as follows:

x y (x,y)
-1 ____
-2 ____
-4 ____
2 ____
8 ____

To find the corresponding values for y, you need to substitute the given x-values into the equation and calculate the corresponding y-values.

e) The objects that might be in the way as you lay the pipe for the sprinkler are not provided in the given information. You would need additional information to determine the potential obstacles.