Math 093 and Math 117A – Linear Functions: Modeling and Applications
1. Advertising Expenses
The total amount of advertising expenses (in billions of dollars) in the United States from 1980 to 1989 is
approximated using the linear model
E = 8.1437 t + 52.982, 0 ≤ t ≤ 9
where E represents the advertising expenses (in billions of dollars) and t represents the year with t = 0
corresponding to 1980.
a) What were the advertising expenses in the year 1983?
b) When were the advertising expenses 118 billion dollars?
c) What is the slope of this line? Interpret within the context of the problem.
d) What is the E-intercept? Interpret within the context of the problem.
e) Interpret in context the meaning of the ordered pair (4, 85.6)
f) Use your knowledge of linear functions to sketch a graph. Label axes in context including units.
g) Interpret in context the t-intercept. Does it make sense?
2. Air Temperature and Altitude
The relationship between the air temperature T (in ºF) and the altitude h (in feet above sea level) is
approximately linear for 0 ≤ h ≤ 20,000. If the temperature at sea level is 60º, an increase of 5000 feet in altitude
lowers the air temperature about 18º.
(a) Write T in terms of h; that is, find the linear function T = mh + b.
(b) Approximate the air temperature at an altitude of 15,000 feet.
(c) Approximate the altitude at which the temperature is 0º. What is the name of this point in the graph?
3. Nutrition.
There are approximately 126 calories in a 2-ounce serving of lean hamburger and approximately 189 calories in
a 3-ounce serving.
a) Write a linear equation for the number of calories in lean hamburger in terms of the size of the serving.
b) Use your equation to estimate the number of calories in a 5-ounce serving of lean hamburger.
c) Interpret in context the ordered pair (10, 630)
4. Olympic Swimming
The winning times in the women’s 400-meter freestyle swimming event in the Olympics from 1948 to 1988 are
modeled by the equation
W = 5.5 – 0.033t, 8 ≤ t ≤ 48
Where y represents the winning time in minutes and t represents the year with t = 0 corresponding to 1940.
a) What was the winning time in the year 1980?
b) When was the winning time 4.44 minutes?
c) Give the value of the W-intercept, and interpret in context.
d) What is the value of the slope? Interpret in context.
e) Find the t-intercept. Interpret in context. Does it make sense?
f) Sketch the graph, label axes in context.
5. Childhood weight
A baby weighs 10 pounds at birth, and three years later the child's weight is 30 pounds. Assume that childhood
weight W (in pounds) is linearly related to age t (in years).
(a) Write W in terms of t; that is, find the linear function W = mt + b.
(b) What is the weight of the child on his/her sixth birthday?
(c) At what age will the child weigh 70 pounds?
(d) Sketch the graph on the interval [0,12]; label in context.
(e) Interpret the t-intercept in context. Does it make sense?
6. Loan repayment
A college student receives an interest-free loan of $8250 from a relative. The student will repay $125 per
month until the loan is paid off.
(a) Express the amount L (in dollars) remaining to be paid in terms of time t (in months); that is, write the linear
function L = mt + b.
(b) After how many months will the student owe $5000?
(c) Find the t-intercept and interpret in context.
(d) Sketch a graph and label in context.
7. Flying Lessons
Flying lessons cost $645 for an 8 hour course and $1425 for a 20 hour course. Both prices include a fixed
insurance fee.
(a) Express the cost, C, of flying lessons in terms of the length, h, of the course; that is, write C = mh + b.
(b) What is the fixed insurance fee? Sketch a graph and label in context.
8. House appreciation
Six years ago a house was purchased for $89,000. This year it was appraised at $125,000. Assume that the
value V of the house after its purchase is a linear function of time t (in years).
(a) Express V in terms of t; that is, write the linear function V = mt + b.
(b) How many years after the purchase date was the house worth $103,000?
(c) Find the x-intercept and interpret in context. Comment on model breakdown if appropriate.
9. Caramel Apple Sales
A vendor has learned that, by pricing caramel apples at $1.25, sales will reach 133 caramel apples per day.
Raising the price to $2.25 will cause the sales to fall to 81 caramel apples per day. Let y be the number of
caramel apples the vendor sells at x dollars each. (a) Write a linear equation that models the number of caramel
apples sold per day when the price is x dollars each. (b) Find the x-intercept and interpret in context.
10. Sales Commissions
A person applying for a sales position is offered alternative salary plans.
Plan A: A base salary of $1,200 per month plus a commission of 4% of the gross sales per month
Plan B: A base salary of $1,400 per month plus a commission of 3% of the gross sales for the month
a. For each plan, write an equation that expresses monthly earnings as a function of gross sales.
b. Graph both equations on the same set of axes. Label in context.
c. For what gross-sales values is plan B preferable?
11. The percentage of Americans owning a car with a cassette tape player during particular years is given in
the following table:
Let t be the number of years since 1980.
(a) Use algebra to find a linear function, f(t), for the line containing the points (5, 29) and (15, 19).
(b) Find the t-intercept. What does it mean in this situation?
(c) Use the linear regression feature of the calculator to find the line of best fit.
12. Grades
A computer science major found that the grade he earned on his programs increased with the time he spent
working on them in the computer lab. He made a table of the hours spent in the lab and the corresponding
program grades. Let g represent the grade earned when h hours are spent in the lab.
a) Sketch a scattergram for the data. Label axes.
b) Sketch a line that comes close to the data points in the scattergram. Use the line to estimate the Program
grade when 3 hours are spent in the lab. State whether you used interpolation or extrapolation to obtain
your result.
c) Does the relation described on the table represent a function? Explain.
d) Write the equation of the line containing the points (1.5, 46) and (5, 98). If necessary, round to one
decimal place.
e) What is the slope of this equation? Interpret within context using correct units.
f) What is the y-intercept of this equation? Interpret in context using correct units.
g) Use the linear regression feature of the calculator to find the line of best fit; round to three decimal
places.
h) Give the correlation coefficient and describe the relationship as weak or strong.
i) Use the linear model from part (g) to predict the Program grade when 3.5 hours are spent in the lab.
j) Find the residual (error) for the estimate found in part (i).
k) Use the equation from part (g) to estimate the number of hours spent in the lab for a student who got a
program grade of 100.
13. Calling Cards
When making a telephone call using a calling card, a call lasting 3 minutes costs $0.80. A call lasting 12
minutes costs $1.70. Let C be the cost, in dollars, of making a call lasting t minutes using a calling card.
a) Write a linear equation that models the cost of making a call lasting t minutes. Use function notation.
b) Find C(8) and interpret in context.
c) Solve C(t) = 1.4 and interpret in context.
d) Interpret in context the mathematical statement C(15) = 2
e) Give the value of the slope; what does it mean in context?
f) Give the value of the y-intercept; what does it mean in context?
14. Arson Arrests
Although the number of people arrested for arson has remained fairly constant during the 1990s, the percent
of arson arrests that have been juveniles has been on the rise. The data is printed in the table below.
a)
b)
c)
d)
e)
f)
g)
(Source: U.S. Justice Department)
Let P represent the percent of arson arrests that are juveniles at t years since 1990. The equation
P(t )  1.83t  44.76 models the data well.
Use the linear model to predict when 75 percent of arson arrests will be juveniles.
Use the linear model to estimate the percent of arson arrests that were juveniles in 1985. Did you
perform interpolation or extrapolation? Explain.
Find the t-intercept. What does it mean in terms of arson arrests?
What is the slope of the equation P = 1.83t + 44.76? What does the slope represent in terms of arson
arrests?
Find the P-intercept. What does it mean in terms of arson arrests?
Find P(10). What does your result mean in terms of the situation?
Find t when P(t) = 100. What does your result mean in term of the situation?
15. The following data represents the height (in inches) and weight (in pounds) of 9 randomly selected
adults.
a) Use the calculator to sketch a scatter-diagram treating height as the independent (explanatory) variable.
b) Use the regression feature of the calculator to find the line of best fit; round to three decimal places.
Match with the shown graph.
c) What is the correlation coefficient? Is the linear relationship strong or weak?
d) Which point, in the scattergraph shown has the largest residual? Find such residual.
e) Use the line to predict the weight of a person who is 64.2 inches tall; round to the nearest pound. Is this
an example of interpolation or extrapolation?
f) Select the correct interpretation of the slope from the following statements.
I.
If height is increased by one inch, then weight will increase by 0.913 pounds
II.
If weight is increased by one pound, then height will increase by 0.913 inches
III.
If height is increased by one inch, then weight will increase by 6.339 pounds
IV.
If weight is increased by one pound, then height will increase by 6.339 inches
Selected Answers
1. a) 77.4 billion dollars, b) In 1988
2. a) T = (-9/2500) h + 60, b) T = 6 ºF, c) h = 16,667 ft.
3. a) y = 63 x, (b) 315 cal. (c) A ten-ounce hamburger has 630 calories.
4. a) 4.18 minutes, (b) In 1972, (c) (0, 5.5) = (0 years since 1940, 5.5 minutes). In 1940, the winning time for
this event was 5.5 minutes. Extrapolation. The model may have been different for the year 1940, (d) m = -0.033
minutes/year since 1940. Every year from 1948 to 1988 the winning time for this event has been decreasing by
0.033 minutes. (e) If y = 0, then t = 167; (167 years since 1940, 0 minutes). In the year 2017 the winning-time
for this event will be 0 minutes which does not make sense; model breakdown has occurred.
5. a) W = (20/3) t + 10, b) 50 lb, c) 9 years old. d) The graph has endpoints at (0,10), and (12,90); (e) if W = 0,
t = -1.5. Model breakdown
6. a) P = -125t +8250, b) 26 months, c) (66 months, 0 dollars), after 66 months the loan will be paid off.
7. (a) C = 65 h + 125; (b) the y intercept is 125; (0,$125). The fixed insurance fee is $125.
8. a) V = 6000t + 89,000, b) 2.3years, (c) (-14.8, 0). It represents the year in which the value of the house was
0 dollars. 14.8 years earlier the price of the house was 0 dollars. It does not make sense; model breakdown.
9. (a) y = -52x + 198; (b) x-intercept: If y = 0, then x = 3.81, ($3.81, 0 caramel apples per day). Consumers will
not buy any caramel apples per day if each costs $3.81.
10. a) Plan A: y = 1200 + .04 x; Plan B: y = 1400 + 0.03x, c) for less than $20,000 in sales.
11. (a) y = -x + 34, (b) ( 34, 0) - In the year 2014, there will be no cars with cassette players, (c) y = -0.823x + 32.619
12. a)
(b)
(b) The Program grade should be about 71. Answers may vary slightly.
Interpolation was used to obtain the result because x = 3 is within the interval [1, 5]
(c) The relation is not a function. Some x-values have two values of y: (2.5, 65), (2.5, 73)
(d) y = 14.9 x + 23.7
(e) m = 14.9. The program grade increases by 14.9 points for every 1-hour increase in the time spent in the lab.
(f) b = 23.3. The model predicts a grade of 23.3 points for a student who spends no hours in the lab.
(g) y = 14.146 x + 28.220
(h) The linear relationship is strong; the correlation coefficient r = 0.975 is close to one
(i) y = 77.7 points
(j) Residual = observed – predicted = 79 – 77.7 = 1.3
(k) x = 5.07 hours
13. a) C(t) = 0.1t + 0.5; (b) C(8) = 1.3. The cost of an 8-minute call is $1.30; (c) If C(t) = 1.4, then t = 9. A 9-minute call
cost $1.40; (d) A 15-minute call costs $2.00; (e) m = 0.1 which means that the cost per minute is $0.10; (f) the y-intercept
b = 0.5 which represents the connection fee. A zero minute call costs $0.50
14. (a) 2007; (b) 36%. Since 1985 is outside the interval [1990, 1995] this is an example of extrapolation. We are not sure
if this model can be used before 1990; (c) (-24.46, 0) No juveniles were arrested for arson in 1965. This does not make
sense. Extrapolation, model breakdown; (d) 1.83; The percent of arson arrests that were juveniles increases by 1.83 each
year; (e) (0, 44.76); In 1990, the percent of arson arrests that were juveniles was 44.76%; (f) 63.06; 63% of arson arrests
were juveniles in 2000; (g) t = 30.19; All arson arrests will be juveniles in 2020. It does not make sense; extrapolation,
model breakdown.
15. (b) y = 6.339 x – 264.107; (c) r = 0.913, strong linear relationship; (d) (67, 185); residual = 185 – 160.6 = 24.4; (e) 143
lb. Interpolation
The following problems do not have answers
1. Biology Biologists 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) What is the slope? Interpret in context.
c) What is the y-intercept? Interpret in context.
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) World record times for the men’s 400-meter run are listed in the table shown below:
Year
1900
1916
1928
1932
1941
1950
1960
1968
1988
1999
Record time
(seconds)
47.8
47.4
47.0
46.2
46.0
45.8
44.9
43.86
43.29
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 in context. Show a rough sketch.
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.
3. Olympic Swimming
The winning times in the women’s 400-meter freestyle swimming event in the Olympics from 1948 to 1988 are
modeled by the equation
y = 5.5 – 0.033t,
8 ≤ t ≤ 48
Where y represents the winning time in minutes and t represents the year with t = 0 corresponding to 1940.
a)
b)
c)
d)
e)
f)
What was the winning time in the year 1980?
When was the winning time 4.44 minutes?
Give the value of the y-intercept, and interpret in context.
What is the value of the slope? Interpret in context.
Find the x-intercept. Interpret. Does it make sense? Comment on model breakdown
Sketch the graph and label.