Math 101 – Applied Problems Using Linear Models
Solve one problem per page.
!!!!!!!NEAT WORK, PLEASE!!!!!!!!
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 y-intercept? Interpret within the context of the problem.
e) Interpret in words the meaning of the ordered pair (4, 85.6)
f) Sketch a graph. Label axes using variable-names and words.
Label any important features.
g) Explain in words the meaning of the x-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 60 º, an
increase of 5000 feet in altitude lowers the air temperature about 18 º.
(a) Express T in terms of h, and sketch the graph.
(b) Approximate the air temperature at an altitude of 15,000 feet.
(c) Approximate the altitude at which the temperature is 0 º . 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 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 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) What was the winning time in the year 1980?
b) When was the winning time 4.44 minutes?
c) Give the value of the y-intercept, and interpret. Does it make sense?
d) What is the value of the slope? Interpret.
e) Find the x-intercept. Interpret. Does it make sense?
f) Sketch the graph and label important features.
1
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) Express W in terms of t.
(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]
(e) Interpret the x(t-)intercept. 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).
(b) After how many months will the student owe $5000?
(c) Find the x-intercept and interpret.
(d) Sketch a graph and label important features.
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. Express the cost, C, of flying lessons in terms of the length, h, of
the course. What is the fixed insurance fee? Sketch a graph and label.
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 .
(b) How many years after the purchase date was the house worth $103,000?
(c) What is the meaning of the x-intercept? Does it make sense within the context of the
problem?
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. Write a linear
equation that models the number of caramel apples sold per day when the price is x dollars
each. What is the meaning of the x-intercept?
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.
c. For what gross-sales values is plan B preferable?
2
Selected Answers to Supplement – Applied Problems Using Linear Models
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. y = 63 x, 315 cal.
4. a) 4.18 minutes, b) In 1972
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)
6. a) P = -125t +8250, b) 26 months, c) 66 months,
7. C = 65 h + 125; $125
8. a) V = 6000t + 89,000, b) 2.3years
9. y = -52x + 198
10. a) Plan A: y = 1200 + .04 x; Plan B: y = 1400 + 0.03x, c) for less than $20,000
3