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Making Connections:Mathematical Modelling
With Exponential andLogarithmic Equations
7.5
Countless freshwater lakes, lush forests, and breathtaking landscapes make
northern Ontario a popular summer vacation destination. Every year, millions
of Ontarians go there to enjoy summer life in the peaceful setting of a cottage,
a campground, or a small town.
Suppose you live and work in northern Ontario as an urban planner. As towns
grow, you will need to pose and solve a variety of problems such as the following.
How much commercial development should be encouraged or permitted?
When and where should a highway off-ramp be built?
Which natural landscapes should be left undisturbed?
These and other related problems may require applying and solving exponential
and logarithmic equations.Careful planning and development can ensure that the natural beauty of our
northern landscape is preserved, while meeting the needs of a growing population.
Take a journey now to Decimal Point, a fictional town located somewhere
in northern Ontario. You have been assigned to perform some urban
planning for this friendly community.
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Example 1 Select and Apply a Mathematical Model
The population of Decimal Point has been steadily growing for several
decades. The table gives the population at 5-year intervals, beginning in
1920, the year the towns population reached 1000.
a) Create a scatter plot to illustrate this growth trend.
b) Construct a quadratic model to fit the data.
c) Construct an exponential model to fit the data.
d) Which model is better, and why?
e) Suppose that it is decided that a recreation centre should be built once the
towns population reaches 5000. When should the recreation centre be built?
Solution
Method 1: Use a Graphing Calculator
a) Clear all equations and Stat Plots from the calculator. Enter the data in
lists L1 and L2 using the list editor.
Turn Plot1 on. From the Zoom menu, choose 9:ZoomStat to display the
scatter plot.
Time (years) Population
0 1000
5 1100
10 1180
15 1250
20 1380
25 1500
30 1600
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b) Use quadratic regression to determine a quadratic equation of best fit,
and store it as a function, Y1, by following these steps:
Presso.
Choose CALC, and then select 5:QuadReg.
PressO 1 for [L1], followed byG.
PressO 2 for [L2], followed byG.
Presss. Cursor over to Y-VARS. Select 1:Function and presse.
The equation of the curve of best fit is approximately
y 0.15x215.4x 1006, where y is the population after x years.
c) To determine an exponential equation of best fit, follow the same stepsas above, except choose 0:ExpReg instead of5:QuadReg. Store the
exponential equation of best fit in Y2.
The equation of the exponential curve of best fit is approximately
P 1006(1.016)t, where P is the population after tyears.
d) Note that both regression analyses yield equations with very high values
ofr2, suggesting that both models fit the given data well. To examine the
scatter plot and both model graphs, pressx to open the graph editor.
Then, ensure that Plot1, Y1, and Y2 are all highlighted. For clarity, the
line style of one of the functions can be altered (e.g., made thick).
Pressf to see how well the two curves fit the given data.
Technology Tip s
Ir2 does not automatically appear:
PressO 0 or [CATALOG]. Pressav to quickly
scroll to the items beginning
with the letter D.
Choose Diagnostics On.
Presse twice.You may need to repeat the
regression step to see r2. This
can be done quickly by using
Oe or [ENTRY]until the regression command
appears, and then pressinge.
C O N N E C T I O N S
Your calculator may display a
value or r2, which is called the
coef cient o determination.
It indicates how close the data
points lie to the curve o themodel. The closer r2 is to 1,
the better the t. You will learn
more about the coef cient o
determination i you study
data management.
C O N N E C T I O N S
The value orshowing on thescreen represents the correlation
coef cient, which measures the
strength and direction o the
relationship betweenxandy.
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It appears that either model fits the data equally well, since the functions
are virtually indistinguishable. Are these models equally valid? Zoom out
to see how the models extrapolate beyond the given data.
Zoom out once:
The models appear to diverge here.
What meaning does the part of the graph to the left of the origin have?
Do you think this a valid part of the domain for this problem? Zoom out
again, and then use the ZoomBox operation to explore this region. Use
the TRACE operation to track the coordinates of each model.
An anomaly occurs when extrapolating the quadratic model back in time.
This model suggests that the population of the town was actually once
larger than it was in year zero, and then decreased and increased again.
This contradicts the given information in the problem, which states that
the towns population had been growing for several decades.
The exponential model gives a more reasonable description of the
population trend before year zero due to its nature of continuous growth.Therefore, the exponential model is better for describing this trend.
Method 2: Use Fathoma) Open a new collection and enter the data into a Case Table.
Technology Tip s
When you pressr, thecursor will trace the points o the
scatter plot, the unctionY1, or
the unctionY2. You can toggle
between these by using the upand down cursor keys. Use the
let and right cursor keys to
trace along a unction graph
or set o points.
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Technology Tip s
To create this New Graph:
Click and drag the graph icon
rom the menu at the top. Click and drag theYear attribute
onto the horizontal axis.
Click and drag thePopulation
attribute onto the vertical axis.
Create a scatter plot ofYear versus Population.
b) Create a dynamic quadratic model by following these steps:
Click and drag three sliders from the menu at the top. Label them a,
b, and c.
Click on the graph. From the Graph menu, choose Plot Function.
Enter the function a*Year^2 b*Year c and click on OK.
Adjust the sliders until a curve of best fit is obtained. Hint: What should
the approximate value ofc be (think about when x 0)?
The quadratic curve of best fit is given approximately by
P 0.15t2 15.5t 1006, where P is the population after tyears.
Technology Tip s
You can adjust the scales o
the sliders by placing the
cursor in various locations and
then clicking and dragging.Experiment with this, noting
the various hand positions that
appear and what they allow
you to do.
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c) An exponential equation can be written in terms of any base. Therefore,
it is possible to determine an equation to model the population, P, of this
town as a function of time, t, in years, in terms of its initial population,
1000, and its doubling period, d:
P 1000 2t_d
Create a dynamic exponential model with a single slider, d. Adjust duntil
the curve of best fit is obtained.
The doubling period is approximately 43.5 years. The exponential
equation of the curve of best fit is approximately P 1000 2t_
43.5.
d) Note that both models fit the data well. To see how well they performfor extrapolation, adjust the axes of each graph.
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e) Use either exponential algebraic model to determine when the recreation
centre should be built for Decimal Point by solving for twhen P 5000.
P 1006(1.016)t
5000 1006(1.16)t
5000_1006
1.016t Divide both sides by 1006.
log (5000_1006) log(1.016)t
log (5000_1006)__log 1.016
t
Apply the power law of
logarithms and divide
both sides by log 1.016.
t 101
P 1000 2t_
43.5
5000 1000 2t_
43.5
Divide both sides by 1000.
5 2t_
43.5
log 5 log (2t_
43.5) Take the common
logarithm of both sides.
log 5 ( t_43.5) log 2 Apply the power law
of logarithms.
Multiply both sides by 43.5(log 5_
log 2
)t 43.5 and divide both sides by log 2.
t 101
Both models indicate that the recreation centre should be built
approximately 101 years after the population of Decimal Point reached
1000. Because the population reached 1000 in 1920, the recreation
centre should be built in the year 2021.
Use a calculator to
evaluate.
Use a calculator to evaluate.
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Example 1 illustrates the important distinction between curve-fitting and modelling.
A well-fit curve may be useful for interpolating a given data set, but such a
model may break down when extrapolated to describe past or future trends.
The town of Decimal Point is enjoying a fiscal surplus, a pleasant situation in
which financial revenues exceed expenses. How should the towns funds beinvested in order to earn the best rate of return?
The compound interest formula modelling the future amount, A, of an
investment with initial principal P is AP(1 i)n, where i is the interest
rate per compounding period, in decimal form, and n is the number of
compounding periods.
Example 2 Investment Optimization
Decimal Point has a surplus of $50 000 to invest to build a recreation centre.
The two best investment options are described in the table.
a) Construct an algebraic model that gives the amount, A, as a function
of time, t, in years, for each investment.
b) Which of these investment options will allow the town to double its
money faster?
c) Illustrate how these relationships compare, graphically.d) If the town needs $80 000 to begin building the recreation centre, how
soon can work begin, and which investment option should be chosen?
Solution
a) Determine the number of compounding periods and the interest rate per
compounding period for each investment. Then, substitute these values
into the algebraic model. Use a table to organize the information.
Lakeland Savings Bond Northern Equity Mutual Fund
Number o compoundingperiods, n
nt n 2t
Interest rate percompounding period, i
61
_4
% per year 0.06256% per year 2 periods per year 0.03
AP(1i)n A 50 000(1.0625)t A 50 000(1.03)2t
Investment Option Lakeland Savings Bond Northern Equity Mutual Fund
Interest Rate 61
_4
% compounded annually 6% compounded semi-annually
Conditions2% of initial principal penalty ifwithdrawn before 10 years
none
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d) The graph indicates that both accounts will reach $80 000 after about
8 years. The Lakeland account earns interest faster, but is it the best
choice for preparing to build the recreation centre? The penalty for
early withdrawal must be considered.
The exponential model can be adjusted for withdrawals that happen
within the first 10 years by subtracting 2% of the initial principal.The adjusted equation becomes
A 50 000(1.0625)t 0.02(50 000)
2% penalty for early withdrawal
or A 50 000(1.0625)t 1000.
Applying a vertical shift to the original amount function can reveal the
effect of this penalty.
The function q(x) represents the adjusted amount function for the
Lakeland account. It is unclear from the graph which account will reach
$80 000 first. Apply algebraic reasoning to decide.
Substitute A 80 000 and solve for t.
Lakeland Savings Bond (penalty adjusted)
A 50 000(1.0625)t 1000
80 000 50 000(1.0625)t 1000
81 000 50 000(1.0625)t
Add 1000 to both sides. 1.62 (1.0625)t Divide both sides by 50 000.
log 1.62 log (1.0625)t Take the common logarithm of both sides.
log 1.62tlog 1.0625
tlog 1.62__
log 1.0625
7.96
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The Lakeland account will reach $80 000 in value after 7.96 years, after
adjusting for the early withdrawal penalty.
Northern Equity Mutual Fund
A 50 000(1.03)2t
80 000
50 000(1.03)2t
1.6 1.032t
log 1.6 log (1.03)2t Take the common logarithm of both sides.
log 1.6 2tlog 1.03
tlog 1.6__
2 log 1.03
7.95
The Northern Equity account will reach $80 000 in value after 7.95 years.
Since the time difference between these two accounts is so small, it does
not really matter which one is chosen, from a purely financial perspective.
Other factors may be considered, such as the additional flexibilityafforded by the Northern Equity account. If the township finds itself in
a deficit situation (where expenses exceed revenues), for example, and if
some of the money in reserve is required for other, more urgent, purposes,
then the Northern Equity account may be preferable.
KEY CONCEPTS
Different technology tools and strategies can be used to construct
mathematical models that describe real situations.
A good mathematical model
is useful for both interpolating and extrapolating from given data in
order to make predictions
can be used, in conjunction with other considerations, to aid in
decision making
Exponential and logarithmic equations often appear in contexts that
involve continuous growth or decay.
Connecting
Problem Solving
Reasoning and Proving
Refecting
Selecting ToolsRepresenting
Communicating
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Communicate Your Understanding
C1 Refer to Example 1. Two regression models were proposed and one
was found to be better.
a) What was the basis for rejecting the quadratic model?
b) Consider a linear model for the data. Is it possible to construct aline that fits the given data reasonably well?
c) Would a linear model be valid for extrapolation purposes?
Explain why or why not.
C2 Explain the difference between curve-fitting and mathematical
modelling. Identify any advantages either procedure has over the other.
C3 Refer to Example 2. Suppose that instead of an early withdrawal
penalty, the investment agency provids a bonus of 2% of the principal
if it is not withdrawn before 10 years have elapsed. How could this be
reflected using a transformation, and when will it apply?
A Practise
For help with questions 1 to 3, refer to Example 1.
1. Plans for Decimal Point call for a highway
off-ramp to be built once the towns population
reaches 6500. When should the off-ramp be built?
2. The town historian is writing a newspaper article
about a time when Decimal Points population
was only 100. Estimate when this was.
3. Refer to the two exponential models developed
in Example 1:
P 1006(1.016)t P 1000 2t_
43.5
a) Use both models to predict
i) the towns population after 100 years
ii) how long it will take for the towns
population to reach 20 000
b) Do these models generate predictions that
are identical, quite close, or completely
different? How would you account for
any discrepancies?
B Connect and Apply
For help with questions 4 and 5, refer to Example 2.
4. Suppose that the
Lakeland Savings
Bond group waives
the early withdrawal
penalty. How mightthis affect the
investment decision for the town?
Provide detailed information.
5. Suppose two other investment options are
available for Decimal Points reserve fund:
Should either of these investments be
considered? Justify your reasoning.
InvestmentOption
Rural OntarioInvestment Group
MuskokaGuaranteedCertifcate
InterestRate
6 1_
2% compoundedsemi-annually
6% compoundedmonthly
Conditions no penalty1% of initial principalpenalty if withdrawnbefore 10 years
Connecting
Problem Solving
Reasoning and Proving
Refecting
Selecting ToolsRepresenting
Communicating
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6. Use Technology The table gives the surface
area of seawater covered by an oil spill as a
function of time.
a) Create a scatter plot of surface area versus
time. Describe the shape of the curve.
b) Perform the following types of regression
to model the data:
i) linear
ii) quadratic
iii) exponential (omit time 0 for this
regression)
Record the equation for the line or curve
of best fit in each case.
c) Assuming that the spill is spreading
isotropically (equally in all directions),
which model do you think makes the
most sense for t 0? Explain why.
d) Use the model that you chose in part c)
to predict
i) the size of the oil spill after 10 min
ii) the length of time it will take for the
spill to reach a diameter of 30 m
e) Describe any assumptions you must make.
7. A $1000 investment earns 8% interest,
compounded quarterly.
a) Write an equation for the value of the
investment as a function of time, in years.
b) Determine the value of the investment after
4 years.
c) How long will it take for the investment to
double in value?
8. Refer to question 7. Suppose that a penalty
for early withdrawal of 5% of the initial
investment is applied if the withdrawal occurs
within the first 4 years.
a) Write an equation for the adjusted value of
the investment as a function of time.
b) Describe the effect this adjustment would
have on the graph of the original function.
9. Use Technology
a) Prepare a cup of
hot liquid, such as
coffee, tea, or hot
water. Carefully
place the cup
on a stable surface in a room at normal
room temperature.
b) Record the temperature of the liquid as it
cools, in a table like the one shown. Collect
several data points.
c) Create a scatter plot of temperature versus
time. Describe the shape of the curve.d) Create the following models for the data,
using regression:
i) quadratic
ii) exponential
Record the equation for each model.
e) Which of these is the better model? Justify
your choice.
f) Use the model that you chose in part e) to
estimate how long it will take for the liquid
to cool to
i) 40C
ii) 30C
iii) 0C
Justify your answers and state any
assumptions you must make.
Time (min) Temperature (C)
0
2
4
Connecting
Problem Solving
Reasoning and Proving
Refecting
Selecting ToolsRepresenting
Communicating
Time (min) Surace Area (m2)
0 01 2
2 4
3 7
4 11
5 14
6 29
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10. Chapter Problem Decimal Point is hosting
Summer-Fest: a large outdoor concert to
celebrate the start of summer. The headline
act is a rising rock group from Australia.
Live, rom Australia:
Koalarox!
Featuring
Rocco Rox on lead guitar!
Boom Boom Bif on drums!
When: July 1, 8:00 p.m.
Where: Integer Island
During sound checks, the bands sound crew
is responsible for setting various acoustic and
electronic instruments to ensure a rich and
balanced sound. The difference in two sound
levels, 1
and 2, in decibels, is given by the
logarithmic equation 2
1 10 log (I2_I
1
), where
I2_I
1
is the ratio of their intensities.
a) Biffs drum kit is miked to produce a sound
level of 150 dB for the outdoor venue. The
maximum output of Roccos normal electric
guitar amplifier is 120 dB. What is the ratio
of the intensities of these instruments?
Explain why Roccos signal needs to beboosted by a concert amplifier.
b) After a few heavier songs, the band plans
to slow things down a bit with a couple of
power ballads. This means that Rocco will
switch to his acoustic guitar, which is only
one ten-thousandth as loud as his normally
amplified electric guitar. By what factor
should the sound crew reduce Biffs drums to
balance them with Roccos acoustic guitar?
Achievement Check
11. Use Technology The table shows the population
growth of rabbits living in a warren.
a) Create a scatter plot of rabbit population
versus time.
b) Perform the following types of regression to
model the data:
i) linear
ii) quadratic
iii) exponential
Record the equation for the line or curve of
best fit in each case.
c) Assuming that the rabbit population had
been steadily growing for several months
before the collection of data, which model
best fits the situation, and why?
d) Use the model to predict when the population
will reach 100.
e) Do you think this trend will continue
indefinitely? Explain why or why not.
Time (months) Number o Rabbits
0 16
1 18
2 21
3 24
4 32
5 37
6 41
7 50
C O N N E C T I O N S
A warren is a den where rabbits live.
C O N N E C T I O N SYou rst compared sound levels using the decibel scale in Chapter 6.
Reer to Section 6.5.
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C Extend and Challenge
12. a) Find some data on the Internet, or elsewhere,
that could be modelled by one or more of
the following:
a line of best fit
a quadratic curve of best fit
an exponential curve of best fit
b) Describe the nature of the data.
c) Use Technology Perform regression analysis
for each type of curve. Record the equation
in each case. How well does each line or
curve fit the data?
d) Which is the best model and why?
e) Pose and solve two problems based on the
data and your best model.13. Use Technology
a) Find some data on the Internet, or elsewhere,
that could be modelled by a logistic curve.
b) Describe the nature of the data.
c) Perform logistical regression analysis.
Record the equation. How well does each
line or curve fit the data?
d) Which is the best model and why?
e) Pose and solve two problems based on the
data and your best model.
14. Use your data from question 13. A piecewise
linear function is a function made up of two
or more connected line segments. Could the
data be modelled using a piecewise linearfunction? If so, do so. If not, explain why not.
15. Math Contest A cyclist rides her bicycle
over a route that is 1_3
uphill, 1_3
level, and 1_3
downhill. If she covers the uphill part of the
route at a rate of 16 km/h, and the level part at
a rate of 24 km/h, what rate would she have to
travel during the downhill part of the route in
order to average 24 km/h for the entire route?
16. Math Contest A circle with radius 2 iscentred at the point (0, 0) on a Cartesianplane. What is the area of the smaller segment
cut from the circle by the chord from (1, 1)
to (1, 1)?
17. Math Contest The quantities x, y, and z are
positive, and xyz_
4. Ifx is increased by 50%,
and y is decreased by 25%, by what percent is
z increased or decreased?
C O N N E C T I O N S
Certain types o growth phenomena ollow a pattern that can be modelled by a logistic unction,
which takes the orm f(x)c__
1aebx, where a, b, and care constants related to the conditions
o the phenomenon, and e is a special irrational number, like . Its value is approximately 2.718.
The logistic curve is sometimes called the S-curve because o its shape.
Logistic unctions occur in diverse areas, such as biology, environmental studies, and business,
in situations where resources or growth are limited and/or where conditions or growth vary over time.
Go to www.mcgrawhill.ca/links/functions12and ollow the links to learn more about logistic unctions
and logistic curves.
y
x0
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