Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.
Exercise 12.2.5
What are the dependent and independent variables?
Answer
dependent variable: fee amount; independent variable: time
Exercise 12.2.6
Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.
Exercise 12.2.7
Graph the equation from Exercise.
Answer
Figure \(\PageIndex{4}\).
Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.
Exercise 12.2.8
Find the equation that expresses the total fee in terms of the number of days the payment is late.
Exercise 12.2.9
Graph the equation from Exercise.
Answer
Figure \(\PageIndex{5}\).
Exercise 12.2.10
Is the equation \(y = 10 + 5x – 3x^{2}\) linear? Why or why not?
Exercise 12.2.11
Which of the following equations are linear?
\(y = 6x + 8\)
\(y + 7 = 3x\)
\(y – x = 8x^{2}\)
\(4y = 8\)
Answer
\(y = 6x + 8\), \(4y = 8\), and \(y + 7 = 3x\) are all linear equations.
Exercise 12.2.12
Does the graph show a linear equation? Why or why not?
Figure \(\PageIndex{6}\).
Table contains real data for the first two decades of AIDS reporting.
Adults and Adolescents only, United States
Year
# AIDS cases diagnosed
# AIDS deaths
Pre-1981
91
29
1981
319
121
1982
1,170
453
1983
3,076
1,482
1984
6,240
3,466
1985
11,776
6,878
1986
19,032
11,987
1987
28,564
16,162
1988
35,447
20,868
1989
42,674
27,591
1990
48,634
31,335
1991
59,660
36,560
1992
78,530
41,055
1993
78,834
44,730
1994
71,874
49,095
1995
68,505
49,456
1996
59,347
38,510
1997
47,149
20,736
1998
38,393
19,005
1999
25,174
18,454
2000
25,522
17,347
2001
25,643
17,402
2002
26,464
16,371
Total
802,118
489,093
Exercise 12.2.13
Use the columns "year" and "# AIDS cases diagnosed. Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?
Answer
The number of AIDS cases depends on the year. Therefore, year becomes the independent variable and the number of AIDS cases is the dependent variable.
Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is \(y = 50 + 100x\).
Exercise 12.2.14
What are the independent and dependent variables?
Exercise 12.2.15
What is the y-intercept and what is the slope? Interpret them using complete sentences.
Answer
The \(y\)-intercept is 50 (\(a = 50\)). At the start of the cleaning, the company charges a one-time fee of $50 (this is when \(x = 0\)). The slope is 100 (\(b = 100\)). For each session, the company charges $100 for each hour they clean.
Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is \(y = 12,000x\).
Exercise 12.2.16
What are the independent and dependent variables?
Exercise 12.2.17
How many pounds of soil does the shoreline lose in a year?
Answer
12,000 pounds of soil
Exercise 12.2.18
What is the \(y\)-intercept? Interpret its meaning.
Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is \(y = 15 – 1.5x\) where \(x\) is the number of hours passed in an eight-hour day of trading.
Exercise 12.2.19
What are the slope and y-intercept? Interpret their meaning.
Answer
The slope is -1.5 (\(b = -1.5\)). This means the stock is losing value at a rate of $1.50 per hour. The \(y\)-intercept is $15 (\(a = 15\)). This means the price of stock before the trading day was $15.
Exercise 12.2.19
If you owned this stock, would you want a positive or negative slope? Why?