8.12: Applications of Exponential Functions
What if you won $500 in a spelling bee competition and invested it into a mutual fund that pays 8% interest compounded annually? How much money would you have after 5 years? After completing this Concept, you'll be able to solve realworld problems like this one that involve exponential functions.
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CK12 Foundation: 0812S Applications of Exponential Functions
Guidance
For her eighth birthday, Shelley’s grandmother gave her a full bag of candy. Shelley counted her candy and found out that there were 160 pieces in the bag. As you might suspect, Shelley loves candy, so she ate half the candy on the first day. Then her mother told her that if she eats it at that rate, the candy will only last one more day—so Shelley devised a clever plan. She will always eat half of the candy that is left in the bag each day. She thinks that this way she can eat candy every day and never run out.
How much candy does Shelley have at the end of the week? Will the candy really last forever?
Let’s make a table of values for this problem.
You can see that if Shelley eats half the candies each day, then by the end of the week she only has 1.25 candies left in her bag.
Let’s write an equation for this exponential function. Using the formula
Now let’s graph this function. The resulting graph is shown below.
So, will Shelley’s candy last forever? We saw that by the end of the week she has 1.25 candies left, so there doesn’t seem to be much hope for that. But if you look at the graph, you’ll see that the graph never really gets to zero. Theoretically there will always be some candy left, but Shelley will be eating very tiny fractions of a candy every day after the first week!
This is a fundamental feature of an exponential decay function. Its values get smaller and smaller but never quite reach zero. In mathematics, we say that the function has an asymptote at
ProblemSolving Strategies
Remember our problemsolving plan from earlier?
 Understand the problem.
 Devise a plan – Translate.
 Carry out the plan – Solve.
 Look – Check and Interpret.
We can use this plan to solve application problems involving exponential functions. Compound interest, loudness of sound, population increase, population decrease or radioactive decay are all applications of exponential functions. In these problems, we’ll use the methods of constructing a table and identifying a pattern to help us devise a plan for solving the problems.
Example A
Suppose $4000 is invested at 6% interest compounded annually. How much money will there be in the bank at the end of 5 years? At the end of 20 years?
Solution
Step 1: Read the problem and summarize the information.
$4000 is invested at 6% interest compounded annually; we want to know how much money we have in five years.
Assign variables:
Let
Let
Step 2: Look for a pattern.
We start with $4000 and each year we add 6% interest to the amount in the bank.
The pattern is that each year we multiply the previous amount by the factor of 1.06.
Let’s fill in a table of values:
We see that at the end of five years we have $5352.90 in the investment account.
Step 3: Find a formula.
We were able to find the amount after 5 years just by following the pattern, but rather than follow that pattern for another 15 years, it’s easier to use it to find a general formula. Since the original investment is multiplied by 1.06 each year, we can use exponential notation. Our formula is
To find the amount after 5 years we plug
To find the amount after 20 years we plug
Step 4: Check.
Looking back over the solution, we see that we obtained the answers to the questions we were asked and the answers make sense.
To check our answers, we can plug some low values of
The answers match the values we found earlier. The amount of increase gets larger each year, and that makes sense because the interest is 6% of an amount that is larger every year.
Example B
In 2002 the population of schoolchildren in a city was 90,000. This population decreases at a rate of 5% each year. What will be the population of school children in year 2010?
Solution
Step 1: Read the problem and summarize the information.
The population is 90,000; the rate of decrease is 5% each year; we want the population after 8 years.
Assign variables:
Let
Let
Step 2: Look for a pattern.
Let’s start in 2002, when the population is 90,000.
The rate of decrease is 5% each year, so the amount in 2003 is 90,000 minus 5% of 90,000, or 95% of 90,000.
The pattern is that for each year we multiply by a factor of 0.95
Let’s fill in a table of values:
Step 3: Find a formula.
Since we multiply by 0.95 every year, our exponential formula is
Step 4: Check.
Looking back over the solution, we see that we answered the question we were asked and that it makes sense. The answer makes sense because the numbers decrease each year as we expected. We can check that the formula is correct by plugging in the values of
Solve RealWorld Problems Involving Exponential Growth
Now we’ll look at some more realworld problems involving exponential functions. We’ll start with situations involving exponential growth.
Example C
The population of a town is estimated to increase by 15% per year. The population today is 20 thousand. Make a graph of the population function and find out what the population will be ten years from now.
Solution
First, we need to write a function that describes the population of the town.
The general form of an exponential function is
Define
Define
Finally we must find what
To find the total population for the following year, we must add the current population to the increase in population. In other words,
The formula that describes this problem is
Now let’s make a table of values.



10  4.9 
5  9.9 
0  20 
5  40.2 
10  80.9 
Now we can graph the function.
Notice that we used negative values of
The question asked in the problem was: what will be the population of the town ten years from now? To find that number, we plug
The town will have 80,911 people ten years from now.
Example D
Peter earned $1500 last summer. If he deposited the money in a bank account that earns 5% interest compounded yearly, how much money will he have after five years?
Solution
This problem deals with interest which is compounded yearly. This means that each year the interest is calculated on the amount of money you have in the bank. That interest is added to the original amount and next year the interest is calculated on this new amount, so you get paid interest on the interest.
Let’s write a function that describes the amount of money in the bank.
The general form of an exponential function is
Define
Define
Now we have to find what
We’re told that the interest is 5% each year, which is 0.05 in decimal form. When we add
The formula that describes this problem is
Solve RealWorld Problems Involving Exponential Decay
Exponential decay problems appear in several application problems. Some examples of these are halflife problems and depreciation problems. Let’s solve an example of each of these problems.
Example E
A radioactive substance has a halflife of one week. In other words, at the end of every week the level of radioactivity is half of its value at the beginning of the week. The initial level of radioactivity is 20 counts per second.
Draw the graph of the amount of radioactivity against time in weeks.
Find the formula that gives the radioactivity in terms of time.
Find the radioactivity left after three weeks.
Solution
Let’s start by making a table of values and then draw the graph.
Time  Radioactivity 

0  20 
1  10 
2  5 
3  2.5 
4  1.25 
5  0.625 
Exponential decay fits the general formula
The formula for this problem is
Watch this video for help with the Examples above.
CK12 Foundation: Applications of Exponential Functions
Vocabulary

General Form of an Exponential Function:
y=A(b)x , whereA= initial value and
Guided Practice
The cost of a new car is $32,000. It depreciates at a rate of 15% per year. This means that it loses 15% of each value each year.
Draw the graph of the car’s value against time in year.
Find the formula that gives the value of the car in terms of time.
Find the value of the car when it is four years old.
Solution
Let’s start by making a table of values. To fill in the values we start with 32,000 at time
Time  Value (thousands) 

0  32 
1  27.2 
2  23.1 
3  19.7 
4  16.7 
5  14.2 
Now draw the graph:
Let’s start with the general formula
In this case:
The formula for this problem is
Finally, to find the value of the car when it is four years old, we plug
Practice
Solve the following problems involving exponential growth.
 Nadia received $200 for her
10th birthday. If she saves it in a bank with a 7.5% interest rate compounded yearly, how much money will she have in the bank by her21st birthday?  Suppose again that Nadia received $200 for her
10th birthday. But what if she saves it in a bank, also with a 7.5% interest rate, but this bank compounds compounds quarterly  how much money will she have in the bank by her21st birthday?  The population of a city grows 15% each year. If the town started with 105 people, how many people will there be in 10 years?

Halflife: Suppose a radioactive substance decays at a rate of 3.5% per hour.
 What percent of the substance is left after 6 hours?
 What percent is left after 12 hours?
 The substance is safe to handle when at least 50% of it has decayed. Make a guess as to how many hours this will take.
 Test your guess. How close were you?

Population decrease: In 1990 a rural area has 1200 bird species.
 If species of birds are becoming extinct at the rate of 1.5% per decade (ten years), how many bird species will be left in the year 2020?
 At that same rate, how many were there in 1980?
 Growth: Janine owns a chain of fast food restaurants that operated 200 stores in 1999. If the rate of increase is 8% annually, how many stores does the restaurant operate in 2007?

Investment: Paul invests $360 in an account that pays 7.25% compounded annually.
 What is the total amount in the account after 12 years?
 If Paul invests an equal amount in an account that pays 5% compounded quarterly (four times a year), what will be the amount in that account after 12 years?
 Which is the better investment?
 The cost of a new ATV (allterrain vehicle) is $7200. It depreciates at 18% per year.
 Draw the graph of the vehicle’s value against time in years.
 Find the formula that gives the value of the ATV in terms of time.
 Find the value of the ATV when it is ten years old.
 The percentage of light visible at
d meters is given by the functionV(d)=0.70d . What is the multiplication factor?
 What is the initial value?
 Find the percentage of light visible at 25 meters.
 A person is infected by a certain bacterial infection. When he goes to the doctor the population of bacteria is 2 million. The doctor prescribes an antibiotic that reduces the bacteria population to
14 of its size each day. Draw the graph of the size of the bacteria population against time in days.
 Find the formula that gives the size of the bacteria population in terms of time.
 Find the size of the bacteria population ten days after the drug was first taken.
 Find the size of the bacteria population after 2 weeks (14 days).
Texas Instruments Resources
In the CK12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9618.
domain
The domain of a function is the set of values for which the function is defined.Exponential Decay Function
An exponential decay function is a specific type of exponential function that has the form , where and .Exponential Form
The exponential form of an expression is , where is the base and is the exponent.Exponential Growth Function
An exponential growth function is a specific type of exponential function that has the form , where and .Range
The range of a function is the set of values for which the function is defined.Image Attributions
Description
Learning Objectives
Here you'll learn how to apply a problemsolving plan to problems involving exponential functions. You'll also learn how to solve realworld applications involving exponential growth and decay.