2.5: Modeling a New Type of Growth

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INTRODUCTION

Throughout this course, we will be discussing two of the most important types of mathematical models: linear models and exponential models.

When a quantity increases or decreases by the same amount for each additional unit of the input variable (usually denoted as x), the value of the output variable (usually denoted as y) can be modeled as a linear model. Linear models have a constant additive rate of change, sometimes called a constant absolute rate of change. For example, think of getting a $500 raise every year. We will discuss this in more detail later in this collaboration.

When a quantity increases or decreases by the same percentage for each additional unit of the input variable (usually denoted as x), the value of the output variable (usually denoted as y) can be modeled as an exponential model. Exponential models have a constant multiplicative rate of change, sometimes called a constant relative rate of change. For example, think of getting a 3% raise every year. We will discuss this in more detail later in this collaboration.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

exponential models are best used to represent quantities which have a constant relative change, often expressed as a percentage or ratio.
the doubling time or halving time associated with an exponential model does not depend on the initial value.

By the end of this collaboration, you should be able to

set up and refine algebraic equations for exponential models in various growth and decay situations.
discuss similarities and differences between linear and exponential models.
use at least one strategy to “solve” an equation in an exponential model.
create exponential models with given half-lives.

PROBLEM SITUATION: EXPONENTIAL MODELS

In Preparation M.5, you looked at some quantities which change over time. Looking back at Preparation M.5 Questions 2 and 3 (shown below for reference), think about the terminology from previous units that might apply to these questions.

Preparation M.5 Question 2: The population of a small town in Ohio was 5,500 people in the year 2010 and census data showed that the population decreased by approximately 2.5% every year until 2020. What was the approximate population in 2011? 2012? 2020? Round to the nearest person.

Preparation M.5 Question 3: The population of a small town in Michigan was also 5,500 people in the year 2010, but it decreased by an average of 150 people a year until 2020. What was the approximate population in 2011? 2012? 2020? Round to the nearest person.

Now apply this terminology to your answer for Preparation M.5 Question 4 (shown below for reference).

(1) Answer Preparation M.5 Question 4 again below by using complete sentences that include the most appropriate terminology. (Hint: Think about the terminology introduced in both Module M and prior Module N.)

Preparation M.5 Question 4: What is different about the population change from year to year in the small town in Michigan (in Question 3 above) and the population change from year to year in the small town in Ohio (in Question 2 above)?

Linear Models (review)

Linear models are based on constant absolute change.

We often say that linear models have a constant additive rate of change. In the context of the Michigan population example (which was increasing by 150 people every year), we would use repeated addition of 150 to calculate the population at any given year.

Exponential Models

When a quantity, such as population, is increasing with a relative multiplicative change each year, this growth can be called constant relative change, since the amount of change at any point in time depends upon the amount of the quantity present at that point in time. This means the ratio of the rate of change of the quantity to its current size remains constant over time. It also means that if the quantity is growing, the more there is of the quantity, the faster it grows.

This type of growth can be modeled with an exponential model. Exponential models are used to represent these situations where a quantity (such as population) increases by a certain factor per unit increase in another quantity (such as time).

As with linear models, exponential models can increase, which is called exponential growth, or decrease, which is called exponential decay.

We often say that exponential models have a constant multiplicative rate of change. Let's consider the Ohio population example, which was decreasing by 2.5% every year, in another way: how to model it if the population were increasing. During Year 1, the population started at 5500 and gained 2.5%, or the population is 5500 + .025 × 5500. We could use the distributive property to write this as 5500(1 + .025), or even better, 5500(1.025).

In other words, if we have a 2.5% increase, we multiply by 1.025. Think of this as you keep 100% of the population (which is why you have the 1) and then gain 2.5% (which is why you have the .025). To find the population at the end of Year 2, you would multiply the population at the end of the 1st year by 1.025, giving 5500(1.025)(1.025), or even better, 5500(1.025)2.

(2) (a) List at least two other situations where quantities, besides population, can have constant relative change and be represented with an exponential model, and explain your reasoning.

Quantity that increases or decreases by the same percentage every day (or year, or month):

Explain your reasoning:

Quantity that increases or decreases by the same percentage every day (or year, or month):

Explain your reasoning:

(b) Refer to the examples from part (a), and create an algebraic formula that represents the changing values over time.

(3) Suppose that the cost of attending a certain four-year private college (tuition and fees) was $16,500 in 1977 and that the cost increased about 3.2% every year.

(a) Create an algebraic model, or formula, that could be used to estimate the cost of attending this college over time.

(b) According to your model, what would be the tuition and fees for 1987, 1997, and 2022? Round your answers to the nearest dollar.

(d) If the original cost of attending the college in 1977 was only $12,000, how would the doubling and tripling time change?

(e) Explain one strategy you could use to determine the year when this model would predict a cost of $42,000. Assume we are using the original cost of $16,500 for 1977.

(4) The rate of elimination of caffeine from the human body varies greatly from individual to individual. Suppose that Jacob drinks a 16-ounce coffee drink that contains about 310 mg of caffeine. His body eliminates about 13% of the caffeine every hour. Hint: Assume all caffeine is absorbed into the body as soon as the coffee is drunk.

(a) Create an algebraic model for this situation. Let t be the number of hours after drinking a 16-oz cup of coffee, and let A be the amount of caffeine in your body.

(b) How much caffeine would be in Jacob’s body right after he drank a second 16-ounce coffee drink, exactly three hours after drinking the first one?

Summary

Throughout this course, we will be discussing two of the most important types of mathematical models: linear models and exponential models.

When a quantity increases or decreases by the same amount for each additional unit of the input variable (usually denoted as x), the value of the output variable (usually denoted as y) can be modeled as a linear model. Linear models have a constant additive rate of change, sometimes called a constant absolute rate of change. Here are some additional examples of exponential models.

Context	Input Variable	Output Variable	Exponential Model
Scott’s starting salary is $55,000, and he will be getting a 2% pay raise every year	Number of years x	Scott’s salary ^y	y=55,0001.02^x
A local Payday-Loan-Cash office charges 25% interest per week on a $1,000 loan	Number of weeks w	Amount owed ^A	A=1,0001.25^w
A brand new Dodge Charger valued at $52,250 depreciates at the rate of 8% per year	Number of years n	Value ^V	V=52,2500.92ⁿ

In general, exponential growth models are written as A=P(1+r)^t, where

P is the starting value
r is the growth rate expressed as a decimal
A is the amount amount after time t

In general, exponential decay models are written as A=P(1-r)^t, where

P is the starting value
r is the elimination rate expressed as a decimal
A is the amount amount after time t

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

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