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. 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.
Suppose you are in the market for a new home and are interested in a new housing community under construction in a different city.
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.
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.
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.
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?
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.
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.
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.
Use elimination to determine how many houses with each floor plan are available. Explain how you arrived at your answer.
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.
Which mover provides a better deal for 2 hours of work? How did you arrive at your answer?
Which mover provides a better deal for 15 hours of work? How did you arrive at your answer?
For what values h (hours) does Quick Move offer the better deal? Express your answer as an inequality. Explain how you reached your answer.
What help do you need. The hints are very detailed.
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. 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?
Gallons of gas.
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.
To answer these questions, we will go step by step and solve each part. Let's begin:
a) The first equation illustrating the situation is:
x + y = 56
b) The second equation illustrating the situation is:
y = 3x
c) To determine how many of each floor plan is available, we'll use substitution. From equation b), we can rewrite it as:
x = (1/3)y
Now, substitute this value of x in equation a):
(1/3)y + y = 56
(4/3)y = 56
Multiply both sides by 3/4:
y = (3/4) * 56
y = 42
Now substitute the value of y back into equation b):
x = (1/3) * 42
x = 14
So, there are 14 houses available with floor plan #1 and 42 houses available with floor plan #2.
d) The intercepts of the equation from part a) represent the points where the line crosses the x and y axes. To find the x-intercept, let y = 0:
x + 0 = 56
x = 56, so the x-intercept is (56, 0).
To find the y-intercept, let x = 0:
0 + y = 56
y = 56, so the y-intercept is (0, 56).
For part b), the intercept is found in the same way:
To find the x-intercept, let y = 0:
x + 0 = 0
x = 0, so the x-intercept is (0, 0).
To find the y-intercept, let x = 0:
0 + y = 0
y = 0, so the y-intercept is (0, 0).
If you solve the system of equations by graphing, the lines will intersect at the point (14, 42).
e) The equation illustrating the situation in the new community is:
x + y = 38
f) The equation illustrating the combined worth of the available houses is:
175,000x + 200,000y = 7,200,000
g) To determine the number of houses with each floor plan available, we can use elimination. Multiply equation e) by -175,000 and equation f) by 38:
-175,000x - 175,000y = -6,650,000
175,000x + 200,000y = 7,200,000
Add the equations together to eliminate x:
25,000y = 550,000
y = 22
Substitute this value of y back into equation e):
x + 22 = 38
x = 16
So, there are 16 houses available with floor plan #1 and 22 houses available with floor plan #2 in the new community.
h) To compare the rates of Heavy Lifters and Quick Move for 2 hours of work, let's calculate the costs for each.
Heavy Lifters: Fee + Hourly rate * hours
Cost = 80 + 35 * 2
Quick Move: Hourly rate * hours
Cost = 55 * 2
Compare the costs to determine which mover provides a better deal.
i) To compare the rates for 15 hours of work, let's calculate the costs for each:
Heavy Lifters: Fee + Hourly rate * hours
Cost = 80 + 35 * 15
Quick Move: Hourly rate * hours
Cost = 55 * 15
Compare the costs to determine which mover provides a better deal.
j) To determine the values of h (hours) for which Quick Move offers the better deal, we need to find when the cost of Quick Move is less than the cost of Heavy Lifters. So we have the inequality:
55h < 80 + 35h
Solve for h:
20h < 80
h < 4
So, for hours less than 4, Quick Move offers a better deal.