# three types of houses

In Metro, there are three types of houses: E (expensive), M (medium), and C (cheap). A recent survey suggest that there are three types of preferences for fire protection: H (high), I (intermediate), and L (low). Mr Wizard, an economic consultant to the city, recently made the following statement: "If my assumptions are correct , the shopping and sorting from the Tiebout process will generate three municipalities"

a). Assume that Mr. Wizard's reasoning is correct. What are his assumptions?

b). Suppose that the equilibrium number of municipalities turn out to be nine, not three. Where did Mr. Wizard go wrong? What set of assumptions would have given him the correction prediction?

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#### Solution Preview

an economic consultant to the city, recently made the following statement:

In Metro, there are three types of houses: E (expensive), M (medium), and C (cheap). A recent survey suggest that there are three types of preferences for fire protection: H (high), I (intermediate), and L (low). Mr Wizard, an economic consultant to the city, recently made the following statement: "If my assumptions are correct , the shopping and sorting from the Tiebout process will generate three municipalities" ...

#### Solution Summary

This solution gives you a detailed discussion on three types of houses.

Landscape Design, Buying a Home and Fueling up

Questions

1. Exercise: Four Concept Check

Post your 50 word response to the following:

Explain in your own words why the line x = 4 is a vertical line.

2. DQ 5-1

Post your response to the following:

What similarities and differences do you see between functions and linear equations?

Are all linear equations functions? Is there an instance when a linear equation is not a function? Support your answer. Create an equation of a nonlinear function and provide two inputs for evaluation.

3. DQ 5-2

Post your response to the following:

What is the difference between domain and range?

Describe a real-life situation that could be modeled by a function.

Provide some feedback and answers. Describe the values for x that may not be appropriate values even when they are defined by function. A function could for example, indicate the amount of bone strength (y) in a living human body over time in years (x). It would not make sense to look at negative years, because the person would not yet be born. Likewise, looking beyond 100 years might not make sense, as many people do not live to be 100.

4. Exercise: Four Concept Check

Post your 50 word response to the following: How can you determine if two lines are

Perpendicular?

5. DQ 7-1

Post your response to the following:

Systems of equations can be solved by graphing or by using substitution or elimination. What are the pros and cons of each method? Which method do you like best? Why? What circumstances would cause you to use a different method?

Consider some of your responses by indicating pros and cons that you may not have considered or persuading them to see the value of the method you like best (if you chose different methods). Describe situations in which you might use their methods of solving.

2. Retail Sales. Mountainside Fleece sold

40 Neckwarmers. Solid-color neckwarmers sold for $9.90 each and print ones sold for $12.75 each. In all, $421.65 was taken in for the neckwarmers. How many of each type were sold?

3. Sales of Pharmaceuticals. In 2004, the Diabetic Express charged $27.06 for a vial of Humulin insulin and $34.39 for a vial of Novolin Velosulin insulin. If a total of $1565.57 was collected for 50 vials of insulin, how many vials of each type were sold?

3. Fundraising. The St. Mark's Community Barbecue served 250 dinners. A child's plate cost $3.50 and an adult's plate cost $7.00. A total of $1347.50 was collected. How many of each type of plate was served?

7. Exercise: Eight Concept Check

Post your 50 word response to the following:

Describe what the graph of interval [-4, 10] looks like.

Problems: 1

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. 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?

Problems: 2

Fueling Up

Motorists often complain about rising gas prices. Some motorists purchase fuel efficient vehicles and participate in trip reduction plans, such as carpooling and using alternative transportation. Other drivers try to drive only when necessary.

Application Practice

Answer the following questions. Click the white space below each question to maintain proper formatting.

1. Suppose you are at the gas station filling your tank with gas. The function C(g) represents the cost C of filling up the gas tank with g gallons. Given the equation:

a) What does the number 3.03 represent?

b) Find C(2)

c) Find C(9)

d) For the average motorist, name one value for g that would be inappropriate for this function's purpose. Explain why you chose the number you did.

e) If you were to graph C(g), what would be an appropriate domain? Range? Explain your reasoning.

2. Examine the rise in gasoline prices from 1997 to 2006. The price of regular unleaded gasoline in January 1997 was $1.26 and in January 2006 the price of regular unleaded gasoline was $2.31 (Bureau of Labor Statistics, 2006). Use the coordinates (1997, 1.26) and (2006, 2.31) to find the slope (or rate of change) between the two points. Describe how you arrived at your answer.

3. The linear equation

represents an estimate of the average cost of gas for year x starting in 1997. The year 1997 would be represented by x = 1, for example, as it is the first year in the study. Similarly, 2005 would be year 9, or x = 9.

a) What year would be represented by x = 4?

b) What x-value represents the year 2018?

c) What is the slope (or rate of change) of this equation?

d) What is the y-intercept?

e) What does the y-intercept represent?

f) Assuming this growth trend continues, what will the price of gasoline be in the year 2018? How did you arrive at your answer?

4. The line

represents an estimate of the average cost of gasoline for each year. The line

estimates the price of gasoline in January of each year (Bureau of Labor Statistics, 2006).

a) Do you expect the lines to be intersecting, parallel, or perpendicular? Explain your reasoning.

b) Use the equations of the lines to determine if they are parallel. What did you find?

c) Did your answer to part b confirm your expectation in part a?

Problems: 3

Buying a Home

For most people, buying a house is a great investment that can offer security in an uncertain world, but buying a house is also a commitment.

Application Practice

Answer the following questions. Click the white space below each question to maintain proper formatting.

1. Suppose you are in the market for a new home and are interested in a new housing community under construction in a different city.

a) The sales representative informs you that there are two floor plans still available, and that there are a total of 56 houses available. Use x to represent floor plan #1 and y to represent floor plan #2. Write an equation that illustrates the situation.

b) The sales representative later indicates that there are 3 times as many homes available with the second floor plan than the first. Write an equation that illustrates this situation. Use the same variables you used in part a.

c) Use the equations from part a and b of this exercise as a system of equations. Use substitution to determine how many of each type of floor plan is available. Describe the steps you used to solve the problem.

d) What are the intercepts of the equation from part a of this problem? What are the intercepts from part b of this problem? Where would the lines intersect if you solved the system by graphing?

2. As you are leaving the community, you notice another new community just down the street. Because you are in the area, you decide to inquire about it.

a) The sales representative here tells you they also have two floor plans available, but they only have 38 homes available. Write an equation that illustrates the situation. Use x and y to denote floor plan #1 and floor plan #2 respectively.

b) The representative tells you that floor plan #1 sells for $175,000 and floor plan #2 sells for $200,000. She also mentions that all the available houses combined are worth $7,200,000. Write an equation that illustrates this situation. Use the same variables you used in part a.

c) Use elimination to determine how many houses with each floor plan are available. Explain how you arrived at your answer.

3. You recently started the paperwork to purchase your new home, and you were just notified that you can move into the house in 2 weeks. You decide to hire a moving company, but are unsure which company to choose. You search online and are interested in contacting two companies, Heavy Lifters and Quick Move, to discuss their rates. Heavy Lifting charges an $80 fee plus $35 per hour. Quick Move charges $55 per hour with no additional fees.

a) Which mover provides a better deal for 2 hours of work? How did you arrive at your answer?

b) Which mover provides a better deal for 15 hours of work? How did you arrive at your answer?

c) For what values h (hours) does Quick Move offer the better deal? Express your answer as an inequality. Explain how you reached your answer.

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