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

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

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?

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?