Precalculus
Lesson- Math Modeling Intro
Name:__________________________________
Date:___________________________________
Objectives:



write polynomial functions to model a situation
determine possible dimensions within a practical domain
determine the area or volume
Def.: Practical Domain:
(1)
A box is to be made out of a piece of sheet metal that
measures 20 cm by 30 cm. Squares, x cm on a side, will be
cut from each corner, and then the ends and sides will be
folded up to create a box that will have no top.
(a)
Write a function to express the volume of the box V(x)
in terms of x.
x
x
x
x
20
cm
x
x
x
x
30
cm
(b)
What is the practical domain?
(c)
Find the dimensions of the box, to the nearest tenth, that would give the maximum volume. What
is the maximum volume to the nearest tenth?
(d)
Sketch the graph and indicate the WINDOW used.
1
(2)
x
x
x
4
4
4
An animal clinic wants to construct a kennel with 3 individual pens,
each with a gate 4 feet wide and an area of 90 square feet. The fencing
does not include the gates.
(a)
Write a function to express the fencing F(x) as a function of x.
(b)
Find the dimensions for each pen, to the nearest tenth of a foot that would produce the required
area of 90 square feet but use the least fencing. What is the minimum fencing to the nearest
tenth?
(c)
Sketch the graph and indicate the WINDOW used.
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(3)
A box is to be made out of a piece of tin that measures 12 cm
by 27 cm. Squares, x cm on a side, will be cut from each
corner, and then the ends and sides will be folded up to create
a box that will have no top.
(a)
Write a function to express the volume of the box V(x)
in terms of x.
x
x
x
x
12
cm
x
x
x
x
27
cm
(b)
What is the practical domain?
(c)
Find the dimensions of the box, to the nearest tenth that would give the maximum volume. What
is the maximum volume to the nearest tenth?
(d)
Sketch the graph and indicate the WINDOW used.
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Precalculus
Lesson: Linear Modeling
Name:______________________________
Date:_______________________________
Objectives:


write linear functions to model a situation
use linear functions to model and determine depreciation
DO NOW: The fixed costs per day for a ski manufacturer are $3,750, and the variable costs are $68 per pair
of skis produced. If x pairs of skis are produced daily, express the daily cost C(x) as a function of x.
(4)
A photo-copier was purchased by a law firm for $8,000 and is assumed to have a salvage value of
$1,000 after 5 years. Its value has depreciated linearly during this period.
(a)
Write a linear function D(t) that relates the value of the photo-copier in dollars to time t in years.
(b)
What would be the depreciated value of the photo-copier after 1 year? 3 years?
(c)
What is the practical domain of the depreciation function?
(d)
Explain what the slope of this linear function represents.
(e)
Sketch the graph and indicate the
WINDOW used.
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(5)
A clothing store marks up the price of a shirt from $20 to $33, and a jacket from $60 to $93. The
markup policy of the store for items that cost over $10 is assumed to be linear.
(a)
Write a linear function R(c) that expresses retail price R in terms of cost c.
(b)
What does this store pay for a suit that retails for $240?
(c)
What is the practical domain of this retail function?
(d)
Explain what the slope of this linear function represents.
(e)
Sketch the graph and indicate the
WINDOW used.
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Precalculus
Lesson: Mathematical Modeling w/ Polynomials
Name:__________________________________
Date:___________________________________
Objectives:
write polynomial functions to model a situation
determine possible dimensions within a practical domain
A freshwater pipeline is to be run from a source on
the edge of a lake to a small resort community on an
island 5 miles offshore. It costs $20,000 per mile to
lay the pipe on land and $32,000 to lay the pipe in
the lake.
(a)
Lake
Island
5 miles
(6)


Write a function to express the total cost C(x)
of constructing the pipeline as a function of x.
Land
Pipe
x
10 – x
10 miles
Freshwater
Source
(b)
What is the practical domain?
(c)
What is the minimum pipeline construction cost to the nearest dollar? How many miles of pipe,
to the nearest tenth, are needed?
(d)
Sketch the graph and indicate the WINDOW used.
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x
(7)
A rectangular area of 60 square meters has a wall on one of its sides, as shown.
The sides perpendicular to the wall are made of fencing that costs $36 per
meter. The side parallel to the wall is made of decorative fencing that costs
$48 per meter.
(a)
Write a function to express the total cost C(x) of the fencing as a function
of x.
W
A
L
L
x
(b)
Find the minimum cost to the nearest dollar. What are the dimensions that give this minimum
cost to the nearest tenth?
(c)
Sketch the graph and indicate the WINDOW used.
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Precalculus
Lesson: Linear and exponential models
Name:______________________________
Date:_______________________________
Objective:
(1)

write linear and exponential functions to model depreciation
At Happy Motors, a car salesperson offers a sale on two types of cars: Car A has a sticker price of
$17,000, but depreciates in value exponentially at an average rate of 20% each year. Car B also has a
sticker price of $17,000, but depreciates to $2,000 in 10 years, which represents a linear depreciation.
(a)
Write an exponential function A(t) that relates the value of Car A in dollars to time t in years.
(b)
Write a linear function B(t) that relates the value of Car B in dollars to time t in years.
(c)
What is the depreciated value of both cars after 3 years?
(d)
In 2004, Rodd decides to buy Car A and Todd decides to buy Car B. Will their cars ever have
the same value again, and if so, during what year(s)?
(e)
Sketch the graphs and indicate the WINDOW used.
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(2)
(3)
Rachel purchased a new stereo system for $2,000 and is assumed to have a depreciated value of $800
after eight years, which represents a linear depreciation during this period. In the same year, her best
friend Joshua bought his new stereo system for $2,500 and is assumed to have an average exponential
depreciation rate of 17.5% per year.
(a)
Write a linear function R(t) that relates the value of Rachel’s stereo in dollars to time t in years.
(b)
Write an exponential function J(t) that relates the value of Joshua’s stereo in dollars to time t in
years.
(c)
What is the depreciated value of each stereo after 1 year?
(d)
Rachel and Joshua both bought their stereos in 2003, but
spent different amounts of money. Will their stereos ever
be the same value again, and if so, during what year(s)?
(e)
Sketch the graphs and indicate the WINDOW used.
Ms. Ingram wants to buy a new computer that will retain its value. She has a choice of buying the
Alpha Computer for $5,000 or the Beta Computer for $4,000. The Alpha Computer is assumed to have
a depreciated value of $1,400 after three years, which represents a linear depreciation. The Beta
Computer depreciates in value exponentially at an average rate of 30% each year.
(a)
Write a linear function A(t) that relates the value of the Alpha Computer in dollars to time t in
years.
(b)
Write an exponential function B(t) that relates the value of the Beta Computer in dollars to time
t in years.
(c)
If Ms. Ingram wants to be able to sell this computer after
4 years, which model will have a higher value?
(d)
Sketch the graphs and indicate the WINDOW used.
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Precalculus
Lesson: Piecewise Models
Name:______________________________
Date:_______________________________
Objective:
(8)

write piecewise functions to model a situation
A beauty supply store sells eyeliner pencils for $4.00 each. For an order of 2 dozen or more pencils,
the price per pencil for all pencils ordered is reduced to $3.50. For an order of 5 dozen or more pencils,
the price per pencil for all pencils ordered is reduced to $3.25.
(a)
Ms. Ingram decides to go shopping and buys the following number of eyeliner pencils. Fill in the
chart to determine the total cost for each order she placed:
Number of
Pencils
10
24
42
59
60
Total
Cost
(b)
Write a piecewise function C(x) that relates the total cost in dollars for an order of x pencils.
(c)
Graph y = C(x) for 0  x  84, and identify any points of discontinuity.
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(9)
On weekends and holidays, Gunning Plumbing’s emergency plumbing repair service charges $2.00 per
minute for the first 30 minutes of a service call and $1.00 per minute for each additional minute.
(a)
Fill in the chart to determine the total cost for each call placed, given the number of minutes:
Time in
minutes
20
42
55
60
73
Total
Cost
(b)
Write a piecewise function P(t) which expresses the total cost of a service call in terms of time t
in minutes.
(c)
Using your function from part (b), how much would a 47 minute service call cost?
call?
A 75 minute
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Precalculus
HW: Piecewise Models
Name:______________________________
Date:_______________________________
Objective:

write piecewise functions to model a situation
(1) In Exponential City, the system for speeding ticket fines is as follows:
Base Fine
$75.00
For speeds 1 – 5 MPH over the posted limit
$5 per MPH
For speeds 5 – 15 MPH over the posted limit
$5 per MPH (for the first five MPH)
$10 per each additional MPH
For speeds over 15 MPH over the posted limit
$5 per MPH (for the first five MPH)
$10 per MPH (for the next ten MPH)
$20 per each additional MPH
(a) Ms. Ingram often commutes on Function Highway, which goes through Exponential City. The
posted speed limit is 55 MPH. Fill in the chart to determine the total cost for each of the
following speeding tickets:
Recorded Speed
Cost of Ticket
58 MPH
61 MPH
70 MPH
50 MPH
(b) Write a piecewise function that relates the total cost in dollars for a speeding ticket in Exponential
City.
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(2) A car rental agency charges $0.35 per mile if the total mileage does not exceed 50. If the total mileage
exceeds 50, the agency charged $0.35 for the first 50 miles plus $0.15 per mile for the additional
mileage. There is an additional charge of $100.00 for the total mileage that exceeds 100 miles.
Write the piecewise function that describes this situation.
(3) For a round trip ticket from Kalamazoo, Michigan to Walla Walla, Washington, a travel agency charged
$89 per person. In order to attract more clients, they offered the lower fare of $82 per person for
bookings of groups of more than 6 people, but less than 11 people. If 11 or more people booked for this
trip, the fare would be offered at $76 per person.
(a) Fill in the chart to determine the total cost for each group, given the number of people booked for
the flight:
People in Group
3
8
15
Total Cost
(b) Write a piecewise function that expresses the total cost of booking this flight.
(c) Graph this piecewise function and label the axes.
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PreCalculus
Review for Test on Function Applications
Name:__________________________________
Date:___________________________________
SHOW ALL WORK ON SEPARATE PAPER!
(1) Carmelo decides to build a dog kennel with 80 feet of fencing wire against the
side of his house, such that one of the lengths of the fence will be replaced by the
wall.
WALL
(a) Write a perimeter (total fencing) equation.
W
(b) Write a function to express the total area A(W) in terms of W.
(c) What is the practical domain?
L
(d) Find the dimensions of W and L, to the nearest tenth, such that the diagram above would have a
maximum area. What is the maximum area to the nearest tenth?
(e) Sketch the graph and indicate the WINDOW used.
(2) A car rental agency charges $0.42 per mile if the total mileage does not exceed 50. If the total mileage
exceeds 50, the agency charged $0.42 for the first 50 miles plus $0.13 per mile for the additional
mileage. There is an additional charge of $100.00 for the total mileage that exceeds 100 miles.
Write the piecewise function that describes this situation.
(3) A digital camera was purchased for $250 and is assumed to have a salvage value of $40 after 7 years.
Its value has depreciated linearly during this period.
(a) Write a linear function V(t) that relates the value of the camera in dollars to time t in years.
(b) What would be the depreciated value of the camera after 2 years? 5 years?
(c) What is the practical domain of the depreciation function?
(d) Sketch the graph and indicate the WINDOW used.
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(4) Jenny wants to buy a new flat screen television that will retain its value. She has a choice of buying the
Luxury Brand television for $3,000 or the Expensive Brand television for $3,500. The Luxury Brand is
assumed to have a depreciated value of $1,200 after three years, which represents a linear depreciation.
The Expensive Brand depreciates in value exponentially at an average rate of 37.5% each year.
(a) Write a linear function L(t) that relates the value of the Luxury Brand television in dollars to
time t in years.
(b) Write an exponential function E(t) that relates the value of the Expensive Brand television in
dollars to time t in years.
(c) If Jenny wants to be able to sell this flat screen television after 5 years, which model will have a
higher value? Explain your answer.
(d) Sketch the graphs and indicate the WINDOW used.
(5) The Great Stride Shoe Company determines that the annual cost of making x pairs of one type of shoe is
$30 per pair plus $100,000 in fixed overhead costs. Each pair of shoes that is manufactured is sold
wholesale for $50.
(a)
Write a linear function C(x) that models the cost of producing x pairs of shoes.
(b)
Write a linear function R(x) that models the revenue produced from selling x pairs of shoes.
(c)
Find how many pairs of shoes must be made and sold in order to break even.
(d)
Sketch the graphs from parts (a) and (b) and indicate the WINDOW used.
(e)
Explain the graphical interpretation of your answer to part (c).
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(6) The amount of tax that a corporation pays depends on its taxable income. The tax table shows the
federal tax rate brackets for various levels of corporate income.
Federal Tax Brackets
Taxable Income ($)
0 – 50,000
50,001 – 75,000
75,001 – 100,000
100,001 – 335,000
Federal Tax Rate (%)
15
25
34
39
(a)
What is the tax bracket for a corporation with a taxable income of $90,000?
(b)
Graph the tax brackets for the different incomes.
(7) Bermuda grass can be found in Africa and Asia, and has an amazing growth rate of 5.9 inches per day.
Suppose you cut a Bermuda grass plant to a length of 2 inches.
(a)
Write a linear function L(x) that models the length of the plant after x days.
(b)
Using your function from part (a), find the length of the plant if you didn’t cut it again for one
week.
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