Lesson 4.2
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).
· 23. Determine the exponential function for a situation when given an initial value and either the growth/decay rate or a second function value. Interpret the initial value and growth rate of an exponential function. Include compound interest as one application.
· 25. Use functional models to make predictions and solve problems.
Specific Objectives
Students will understand
· the differences and similarities between exponential growth and decay.
· the differences between linear and exponential growth
Students will be able to
· write an equation for an exponential decay model.
In this lesson, you will connect the exponential mathematics of compounding to related applications, such as automobile depreciation and spread of disease.
Depreciation
Problem Situation 1: Understanding Depreciation
Depreciation is a process of losing value, opposite to that of accruing interest. For example, new automobiles lose 15% to 20% of their value each year for the first few years you own them.
Suppose you purchase a a $26,000 automobile that depreciates 15% per year.
#1 Points possible: 5. Total attempts: 5
What will the value of the car be when it is 1 year old?
$
#2 Points possible: 5. Total attempts: 5
Think back to the last lesson, and how you came up with the general formula for the balance of the 5year CD after t years.
Using the same idea, find a formula for the depreciated value (V) of this car after t years.
V =
Hint: If you get stuck, we’ll offer some hints
#3 Points possible: 5. Total attempts: 5
Using the formula you just created, predict the value of the car after 5 years. Round to the nearest cent if needed.
$
#4 Points possible: 5. Total attempts: 5
If you were to graph the value of the car over time, which of the following graphs reflects what you would expect to see?
In the last lesson, you looked at compound interest which is an example of exponential growth. In the previous problem, depreciation is an example of exponential decay. Note the similarities and differences to the growth model.
· Both have a vertical intercept that represents the starting value and is in the equation.
· The base of the exponent in the growth model is >1. The base of the exponent in the decay model is between zero and one.
· Both the compound interest formula (exponential growth) and the decay model have the same basic form, y=Caxy=Cax , with a starting value (C) that is multiplied by a factor raised to a variable power. The base of the exponential is 1+growth rate1+growth rate for growth, and 1−decay rate1decay rate for decay. Note that the general compound interest formula looks different, but the (1+rk)(1+rk) factor can be simplified down to one number.
· In growth, rate of change starts slow and increases. In decay situations, the rate of change initially represents a steep negative decline but becomes less steep (but still negative) as time goes on.
· Both are based on multiplicative (or relative) change.
Spreading Disease
Problem Situation 2: A Spreading Disease
During 2014, there was an Ebola breakout in West Africa. On April 1, 2014, there had been 130 cases reported. A month later there had been 234 cases reported.
#5 Points possible: 12. Total attempts: 5
Find the absolute and relative change between the two reported values.
Absolute change:
Relative change: As a decimal: . As a percent: %
#6 Points possible: 5. Total attempts: 5
Let C represent the number of cases after t months, so t = 0 corresponds to April 2014.
Find a formula for a linear model based on the data given, assuming the trend continues.
C =
Hint: If you get stuck, we’ll offer some hints
#7 Points possible: 5. Total attempts: 5
Let C represent the number of cases after t months, so t = 0 corresponds to April 2014.
The relative change you found two questions earlier is the percent increase, or percent growth rate, during the month. Use it to find a formula for an exponential model based on the data given, assuming the trend continues.
C =
Hint: If you get stuck, we’ll offer some hints
#8 Points possible: 10. Total attempts: 5
What does each model predict the number of cases to be after 6 months? Round to the nearest whole number.
Linear model: cases
Exponential model: cases
#9 Points possible: 10. Total attempts: 5
What does each model predict the number of cases to be after 2 years (24 months)? Round to the nearest whole number.
Linear model: cases
Exponential model: cases
Summary of Exponential Equations
In this lesson and the previous one you learned about exponential models. Exponential equations are used when a quantity is growing or decreasing by the same percent every time interval.
Exponential Equations
Exponential equations have the form y=P(1+r)ty=P(1+r)t
· y is the output variable
· P is the initial value, or vertical intercept (yintercept)
· r is the percent growth rate (relative growth rate) written as a decimal
· If the quantity is growing, r will be positive
· If the quantity is decreasing, then r will be negative
· t is time
A special case of the exponential equation is the compound interest equation, which looks like A=P(1+rn)ntA=P(1+rn)nt , where n is the number of compounds in a year.
To find an exponential equation from given information:
· The initial value will be given. Identify it
· Is the percent growth rate given?
· If yes, identify it
· If no, then look for a second output value, and find the relative (percent) change
Example 1: A population is currently 30,000 and is growing by 4.2% per year.
· Initial population is P=30000
· Growth rate is r= 4.2% = 0.042
· Equation will be y=30000(1+.042)t=30000(1.042)ty=30000(1+.042)t=30000(1.042)t
Example 2: A car was purchased for $12000 and its value decreases by 8% each year
· Initial value is P=12000
· Growth rate is r= 8% = 0.08. Notice it is negative because it’s decreasing in value
· Equation will be y=12000(1−0.08)t=12000(0.92)ty=12000(10.08)t=12000(0.92)t
Example 3: A biologist started with 200 bacteria in a dish, and in 1 hour the amount grew to 300. Find an exponential equation for the number of bacteria.
· Initial value is P=200
· We don’t know the growth rate, so we’ll find the relative change from 200 to 300: 300−200200=100200=0.5300200200=100200=0.5 . So the growth rate is 50% = 0.5.
· Equation will be y=200(1+0.5)t=200(1.5)ty=200(1+0.5)t=200(1.5)t , where t is in hours.
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· Cars lose value quickly after purchase.
· Multiplying a number by a second number between 0 and 1 will give you a larger number than you started with.
· All exponential models share certain characteristics such as the general shape of the graph, increase or decrease by a percentage, and having a vertical intercept.
· Depreciation can be modeled with an exponential equation.
#2 Points possible: 8. Total attempts: 5
One way to describe a linear model is that it is based on additive change while an exponential model is based on multiplicative change. Complete the sentences below illustrating why these terms apply to each model.
In the linear table below, each time x increases by 1, we .
In the exponential table below, each time x increases by 1, we .
y = 2x+1

y = 3(2)x

#3 Points possible: 12. Total attempts: 5
Select the letter of each graph next to the equation that best matches it.
y = 1,000(0.95)x y = 200 + 11x y = 5(1.1)x
#4 Points possible: 15. Total attempts: 5
Certain drugs are eliminated from the bloodstream at an exponential rate. Doctors and pharmacists need to know how long it takes for a drug to reach a certain level to determine how often patients should take medications. Answer the following questions about this situation.
a. Select the correct statement:
· More of the drug will be eliminated in the first hour than in the second hour.
· Less of the drug will be eliminated in the first hour than in the second hour.
· The same amount of the drug will be eliminated in the first hour as in the second hour.
b. Write an exponential equation for the following situation. The drug dosage is 500 mg. The drug is eliminated at a rate of 5.2% per hour. Use D = the amount of the drug in milligrams and t = time in hours.
c. How much of the drug is left after 6 hours? Round to the nearest milligram. milligrams
#5 Points possible: 18. Total attempts: 5
In Lesson 1.2, you looked at historical data of the world’s population. Scientists use models to project population in the future. While these models have limitations, they help leaders plan ahead for the resources people will need. The World Bank estimates that that the world population growth rate in 2010 was 1.1%. The U.S. Census Bureau estimated the world population in 2010 to be about 6.9 billion people.2
a. This is an exponential situation because the population is increasing each year by a percentage of the previous year. Complete the table below based on the growth rate of 1.1%. Some of the entries are done for you. Round projected populations to 2 decimal places.
Calendar Year  Number of Years after 2010  Projected Population 
2010  0  6.90 billion 
2011  1  billion 
2012  billion  
2013  billion 
b.
c. Write an exponential equation for the world population growth after 2010. Let P = the projected population in billions and t = the number of years after 2010. Hint: Think about how you calculated the entries in the table. If you started with 6.9 billion each time, how could you calculate each population? Think back to your work in the lesson.
d. Use your model to predict the population in 2020. billion
#6 Points possible: 10. Total attempts: 5
The Center for Disease Control (CDC) published the following information about respiratory disease in children. Respiratory disease is an illness that affects a person’s ability to breathe and use oxygen.3
In 2005, approximately one fourth of the 2.4 million hospitalizations for children aged < 15 years were for respiratory diseases, the largest category of hospitalization diagnoses in this age group. Of these, 31% were for pneumonia, 25% for asthma, 25% for acute bronchitis and bronchiolitis, and 19% for other respiratory diseases, including croup and chronic disease of tonsils and adenoids.
a. Based on this information, how many children were hospitalized for pneumonia in 2005? children
b. Which of the following pie charts accurately represents the data for children hospitalized for respiratory diseases?
#7 Points possible: 15. Total attempts: 5
A ball is placed at the top of ramp and released. It’s height above ground over time is shown in the table below.
Time (seconds)  Height (cm) 
0  10 
1  9 
2  6 
3  1 
a. Plot the points from the table.
Height (cm)  Clear All Draw: 
Time (sec) 
b.
c. Is this a linear model? Why or why not?
Is this an exponential model? Why or why not?