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. 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?
I don't intend to do this assignment for you.
How about letting us know what answers you have so far, and somebody will give you corrections.
(x+6)(x^2+3x-4)
a) The inequality demonstrating how much money you are willing to spend on the landscaping project can be written as:
$6,000 ≤ x ≤ $10,000
This means that the amount of money you are willing to spend, represented by x, must be between $6,000 and $10,000.
b) To find the maximum number of trees you can buy with a budget of $2,500 for rock and trees, you need to set up an inequality. Let's assume the number of trees you can buy is represented by t. The cost of the rock is given by the equation 30 tons * $60/ton = $1800.
So, the inequality can be written as:
$60t + $1800 ≤ $2500
To solve this inequality, we subtract $1800 from both sides:
$60t ≤ $700
And then divide both sides by $60:
t ≤ $700 / $60
t ≤ 11.67
Since you cannot buy a fraction of a tree, the maximum number of trees you can buy is 11.
c) No, 5 trees would not be a solution to the inequality in part b because 5 exceeds the maximum number of trees that can be bought within the budget of $2,500. In part b, we found that the maximum number of trees is 11.
2. For this part of the question, I cannot directly provide the coordinates of Tree A, Plant B, Plant C, Patio D, Plant E, and Plant F as I am an AI text-based interface and cannot see or interpret images or graphs. However, if you have a coordinate graph of the backyard, you can identify the coordinates by locating the specific points on the graph. The coordinates would typically be represented as ordered pairs (x, y), where x refers to the horizontal position and y refers to the vertical position of the point.
b) To determine whether you can place the pink flamingo lawn ornament at the point (-4, -10), you need to see if this point lies on or above the water line. If the y-coordinate of the point (-4, -10) is greater than or equal to the y-coordinate of the water line equation, then it is safe to place the ornament.
c) To find the slope and y-intercept of the water line equation, you need to have the equation of the line. Unfortunately, in the given information, the equation of the water line is missing. Without the equation of the line, we cannot determine its slope or y-intercept.
d) To complete the table for the sprinkler line equation, you need to substitute different x-values into the equation and calculate the corresponding y-values. The equation of the sprinkler line is missing in the given information, so I cannot provide the table without that information.
e) Without the information about the sprinkler line equation or any obstacles or objects in the backyard, it is not possible to determine what objects might be in the way as you lay the pipe for the sprinkler.