Math 101, 103 – Linear Functions: Modeling and Applications
Steps
Display the given information in a T table. Label variables with names and units.
(You will be given enough information for two points on the line)
Find the equation of the line going through two points.
(First find the slope, then find the equation)
Use the equation to answer the questions
1. Jury Awards in Medical Malpractice. The average jury awards in medical malpractice were 1.2 million dollars in 1994, and 3.4 million dollars in 1999.
a) Assuming they follow a linear pattern, find an equation for the average amount of jury award A (in millions of dollars) as a function of the year t since 1990.
b) Sketch a graph. Label axes using variable names and units.
c) What is the y-intercept? What is the meaning within the context of the problem? Does it make sense?
d) What is the slope of this line? What is the meaning within the context of the problem?
e) What is the x-intercept of this equation? What does it represent? Does it make sense?
2. Telecommunications. A cellular phone company offers several different options for using a cellular telephone. One option, for people who plan on using the phone only in emergencies, costs the user $4.95 per month plus $.59 per minute for each minute the phone is used. Write a linear equation for the monthly cost of the phone in terms of the number of minutes the phone is used. Use your equation to find the monthly cost of using the cellular phone for 13 min in one month.
3. Height of a Mountain Climber. A mountain climber is climbing up a 500-foot cliff. By 1 P.M., he has climbed 115 feet up the cliff. By 4 P.M. he has reached a height of 280 feet.
a) Find an equation that relates the height of the climber to the time measured in hours after 12 noon.
b) Estimate the time when he will reach the top of the cliff.
Math 101, 103 – Scatter Diagrams – Finding an Equation for Linearly Related Data –
Instructions for the TI 83
Enter data
Enter (x) into List 1
Enter (y) into List 2
To Edit a List
Press STAT
Select 1:Edit by pressing ENTER
Arrow up to “sit” on the name of the list, press CLEAR, ENTER to clear the
List (if necessary)
Enter data by pressing ENTER after each number
2nd QUIT to exit the editor and get to the “home screen”
Construct a Scatter plot
Press 2nd STAT PLOT and press ENTER (this selects Plot 1)
Press ENTER to turn the plot on
Arrow to the first graph icon and press ENTER (this selects a scatter plot)
Arrow down, select L1 for Xlist, and press ENTER
(L1 is the 2ndfunction of the key for number 1)
Arrow down, select L2 for Ylist, and press ENTER
(L2 is the 2nd function of the key for number 2)
Press ZOOM 9 (this opens window to see scatter plot)
Press TRACE and arrow right/left (this shows the ordered pairs)
Find the Regression Equation
To calculate the regression equation, in the Home Screen, and to paste it into Y1:
Follow the steps outlined below until you have a command like this in the home screen of your calculator: LinReg(ax+b) L1,L2,Y1
Press STAT, arrow to CALC, and select 4:LinReg(ax+b)
Select L1, L2,
To get Y1 press VARS, select Y-VARS, select Function, and then Y1
Press ENTER
To Graph the line along with the scatter graph
Press ZOOM 9:Zoom Stat
To find the predicted values of y
2nd TRACE (CALC)
Select 1:value
Type the given x value and ENTER
Note: This feature works only if the x-value is between the xmin and xmax of your window. You may want to change those values if necessary, or you may evaluate using the TABLE (ASK Mode)
Math 101, 103 – Using Lines to Model Data – Finding the Line of Best Fit
4) Problem 2, page 63, Lehmann’s Book
The average jury awards in medical malpractice suits have increased greatly in the past decade (see table).As a result, malpractice insurance rates have increased, forcing some doctors to quit practicing or to practice medicine defensively, ordering extra tests or choosing procedures that limit their risks.
year / Average jury award (millions of dollars)1994 / 1.2
1996 / 1.9
1998 / 3.0
1999 / 3.4
2000 / 3.5
a) Use your calculator to sketch a scattergram of the data. Let t be the year since 1990, and A be the average jury award (in millions of dollars). Label axes using variable names and units.
b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places.
c) What is the y-intercept of this equation? What is the meaning within the context of the problem?
d) What is the slope of this equation? What is the meaning within the context of the problem?
e) Find A(5). What is the meaning within the context of the problem? Did you perform interpolation or extrapolation to obtain your result?
f) If this pattern continues, what will be the average jury award in the year 2005?
Are you performing interpolation or extrapolation?
g) Use your model to predict in which year was the average jury award 2.4 million dollars.
5) Olympics
The following table gives the winning Olympic pole vaults in this century until 1988.
Year / Height (ft)1952 / 14.92
1956 / 14.96
1960 / 15.42
1964 / 16.73
1968 / 17.71
1972 / 18.04
1976 / 18.04
1980 / 18.96
1984 / 18.85
1988 / 19.77
1992 / 19.02
1996 / 19.42
From the table we see that athletes generally pole-vault higher as time goes on.
Let H represent the record height in feet, at t years since 1900
a) Use your calculator to sketch a scattergram of the data. Label axes using variable names and units.
b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places.
c) What is the y-intercept of this equation? What is the meaning within the context of the problem?
d) What is the slope of this equation? What is the meaning within the context of the problem?
e) What is the x-intercept of this equation? What is the meaning within the context of the problem? Comment on model breakdown.
f) Find t if H(t) = 14.15 ft. Interpret your results in words.
g) Find H(104) and interpret within the context of the problem. Look up the actual record and compare.
h) Make up a problem in which you have to perform extrapolation to obtain your answer. Comment on the results.
Math 101, 103 – Linear Functions: Modeling and Applications
Steps
Display the given information in a T table. Label variables with names and units.
(You will be given enough information for two points on the line)
Find the equation of the line going through two points.
(First find the slope, then find the equation)
Use the equation to answer the questions
1. Records in the 400-meter run In 1932, the record for the 400-meter run was 46.2 seconds. In 1960, it was 44.9 seconds. Let R represent the record in the 400-meter run and t the number of years since 1900.
a) Write a linear equation relating R and t.
b) Use this equation to predict what the record was is 1988 and 1999.
c) When will the record be 40 seconds?
d) Find the x(t-)-intercept. Interpret. Does it make sense?
2. Temperature scalesThe freezing point of water is 0 ºC, or 32 ºF, and the boiling point is 100 ºC, or 212 ºF.
(a) Express the Fahrenheit temperature F as a linear function of the Celsius temperature C.
(b) What is the Fahrenheit reading of 40 ºC?
(c) What is the x-intercept? (C-intercept). Interpret within the context of the problem.
(d) Now express the Celsius temperature in terms of Fahrenheit temperature.
3. Computers On the basis of data from the US Department of Commerce, there were 24 million homes with computers in 1991. The average rate of growth in computers in homes is expected to increase by 2.4 million homes per year through 2005.
a) Write a linear equation for the number of homes with computers in terms of the year.
b) Sketch a graph. Label axes using variable names and units.
c) Use your equation to find the number of homes that had computers in 2001.
d) Interpret the ordered pair (1995, 33.6)
Math 101, 103 – Using Lines to Model Data – Finding the Line of Best Fit
4) Problem 14, page 73, Lehmann’s Book
World record times for the men’s 400-meter run are listed in the table shown below:
Year / Record time (seconds)1900 / 47.8
1916 / 47.4
1928 / 47.0
1932 / 46.2
1941 / 46.0
1950 / 45.8
1960 / 44.9
1968 / 43.86
1988 / 43.29
1999 / 43.18
Let R represent the record time (in seconds) at t years since 1900
a) Use your calculator to sketch a scattergram of the data. Label axes using variable names and units.
b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places.
c) What is the y-intercept of this equation? What is the meaning within the context of the problem?
d) What is the slope of this equation? What is the meaning within the context of the problem?
e) What is the x-intercept of this equation? What is the meaning within the context of the problem? Comment on model breakdown.
f) Find t if R(t) = 42 seconds. Interpret your results in words.
g) Find R(36) and interpret within the context of the problem.
h) Make up a problem in which you have to perform extrapolation to obtain your answer. Comment on the results.
5) When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The following table lists the results of several such experiments by different scientists.
Asbestos Exposure (fibers/mL) / Percent that develop lung tumors50 / 2
400 / 6
500 / 5
900 / 10
1100 / 26
1600 / 42
1800 / 37
2000 / 28
3000 / 50
a) Use your calculator to sketch a scattergram of the data. Label axes using variable names and units.
b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places.
c) What is the y-intercept of this equation? What is the meaning within the context of the problem?
d) What is the slope of this equation? What is the meaning within the context of the problem?
e) Does it make sense to find x when y = 104? Explain.
g) Find f(2500) and interpret within the context of the problem.
Math 101, 103 – Workshop – Linear Functions: Modeling and Applications
Steps
Display the given information in a T table. Label variables with names and units.
(You will be given enough information for two points on the line)
Find the equation of the line going through two points.
(First find the slope, then find the equation)
Use the equation to answer the questions
1. BiologyBiologists have observed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 120 chirps per minute at 70 °F and 168 chirps per minute at 80°F.
a) Find the linear equation that relates the temperature t and the number of chirps per minute N.
b) Interpret the slope and the y-intercept.
c) If the crickets are chirping at 150 chirps per minute, estimate the temperature.
d) How many chirps per minute at 90°F?
e) Sketch a graph and label.
2. Wages. A manufacturer pays its assembly line workers $11.50 per hour. In addition, workers receive a piecework rate of $0.75 per unit produced.
a) Write a linear equation for the hourly wages W in terms of the number of units x, produced per hour.
b) How many units are produced per hour if the hourly wage is $19?
3. Cell Phone Rates. Suppose you have a cellular phone with a rate plan that costs $20 per month, with no additional charge for the first 30 minutes of use, and then $0.40 for each minute or fraction of a minute after the first 30 minutes.
a) Write an equation that gives the monthly cost C to talk for t minutes, if t is less than 30 minutes.
b) Write an equation that gives the monthly cost C to talk for t minutes, if t is greater than 30 minutes.
Math 101, 103 – Using Lines to Model Data – Finding the Line of Best Fit
4) The table lists average carbon dioxide (CO2) levels in the atmosphere, measured in parts per million (ppm) at Mauna Loa Observatory from 1980 to 1996.
Year / CO2 level (ppm)1980 / 338.5
1982 / 341.0
1984 / 344.3
1986 / 347.0
1988 / 351.3
1990 / 354.0
1992 / 356.3
1994 / 358.9
1996 / 362.7
Let L represent CO2 level at t years since 1980
a) Use your calculator to sketch a scattergram of the data. Label axes using variable names and units.
b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places.
c) What is the y-intercept of this equation? What is the meaning within the context of the problem?
d) What is the slope of this equation? What is the meaning within the context of the problem?
e) What is the x-intercept of this equation? What is the meaning within the context of the problem? Comment on model breakdown.
f) Find t if C(t) = 400 ppm. Interpret your results in words. Comment on extrapolation.
g) Find C(18) and interpret within the context of the problem.
h) The actual CO2 level for the year 1998 was 366.7 ppm.
What is the error of the estimate found in part (g).
5) Biologists have observed that the chirping rate of crickets of a certain species is related to temperature. The table shows the chirping rates for various temperatures.
Temperature °F / Chirping rate (chirps/minutes)50 / 20
55 / 46
60 / 79
65 / 91
70 / 113
75 / 140
80 / 173
85 / 198
90 / 211
Let R represent the chirping rate at a temperature T.
a) Use your calculator to sketch a scattergram of the data. Label axes using variable names and units.
b) Use the calculator to find the line of best fit. Write the equation using 3 decimal places.
c) What is the y-intercept of this equation? What is the meaning within the context of the problem?
d) What is the slope of this equation? What is the meaning within the context of the problem?
e) What is the x-intercept of this equation? What is the meaning within the context of the problem? Comment on model breakdown.
f) Find T if R(T) = 150. Interpret your results in words.
g) Find R(90) and interpret within the context of the problem.
h) Make up a problem in which you have to perform extrapolation to obtain your answer. Comment on the results.