section 7.5 page 393 to 407

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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.


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


Refecting


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


Refecting


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


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?


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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