EL-3 Growth and Decay Models

Section EL-3 Growth and Decay Models

Worksheet Exponential Models

For this outcome Specifically, for this outcome, students should be able to:

Use given information to setup and solve a compound interest problem.
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Use given information to setup and solve a continuous compound interest problem.
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Use given information to setup and solve an exponential growth or decay problem.
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Compound Interest

Compound interest is calculated using the formula $A = P(1 + r)^t\text{,}$ where $A$ is the amount of money accumulated after $t$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (in decimal form), and $t$ is the time the money is invested for in years.

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On quizzes and assessments, you will be expected to solve these problems without a calculator, so you should be able to setup and simplify the expressions to get simplified answers.

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Continuous Compound Interest

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Continuous compound interest is calculated using the formula $A = Pe^{rt}\text{,}$ where $A$ is the amount of money accumulated after $t$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (in decimal form), and $t$ is the time the money is invested for in years.

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On quizzes and assessments, you will be expected to solve these problems without a calculator, so you should be able to setup and simplify the expressions to get simplified answers.

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Exponential Growth and Decay

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Exponential growth and decay are modeled by the formula $A = A_0 a^{t}\text{,}$ where $A_0$ is the initial amount, $a$ is the growth or decay constant, and $t$ is time. If $a \gt 1\text{,}$ the function represents exponential growth; if $0 \lt a \lt 1\text{,}$ it represents exponential decay.

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For any growth or decay model, you should be able to set up the model, solve for the unknown variable (such as time, rate, or initial amount), and interpret the results. You will not be allowed to use a calculator on the quizzes or the assessments,so you should be comfortable with solving these problems without a calculator, to get simplified expressions.

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Worksheet EL-3 Modeling with Exponential Functions

Compound Interest

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Continuous Compound Interest

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Growth and Decay Models

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Section EL-3 Rubric

Worksheet Worksheet

Before attempting the EL-3 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.

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Table 21. EL3 Rubric

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Criteria	M	P	R	N
Simplifying Algebraic Expressions	All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner	No algebraic errors OR some expressions are not simplified.	There are some algebraic errors and some expressions were not simplified.	There are some algebraic errors and some expressions were not simplified.
Problem Structure and Mathematical Communication	Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution.	Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution.	Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution.	There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
Setting up and simplifying solutions to interest problems.	Given information about interest should be able to correctly determine between compound interest and continuous compound interest. Should be able to set up the function with correct principal, rate, time and/or compounding factor. Should be able to simplify the expression.	Given information about interest should be able to determine between compound interest and continuous compound interest. Should be able to set up the function with the principal, rate, time and/or compounding factor.	Errors in setting up the interest function, and/or not able to fully simplify the expression.	No evidence of underrstanding the interest type or the formula needed to solve
Setting up exponential growth and decay problems.	Given information about an exponential model, determine if the function is exponential growth or decay; correctly find the growth rate, and initial value; and give the function in the requested form.	Given information about an exponential model, determine if the function is exponential growth or decay; find a growth rate, and an initial value; and give the function in the requested form.	Set up an exponential model but not in requested form. May have errors in the growth rate and/or the initial value.	No evidence of understanding what an exponential growth or decay model is or how to find the initial information for the model.
Solving exponential growth and decay problems.	Given a specific time, find the amount of the model at that time (answers should be simplified expressions). Given an amount, find the time needed to reach that amount (answers should be simplified expressions in terms of natural logarithms).	Given a specific time, find the amount of the model at that time with minor errors. Given an amount, find the time needed to reach that amount with minor errors.	Incorrect methods used to solve for the time or the amount OR decimal answers were given with little algebraic work leading to a solution.	No evidence shown of finding the amount or the time needed in the exponential model.

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Worksheet EL-3 Modeling Exponential Functions

This outcome covers modeling with exponential functions. You should be able to create exponential models for real-world scenarios and use them to make predictions. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.

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For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:

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Objectives

Simplifying Expressions: Simplify algebraic expressions by combining like terms and applying the distributive property. Answers should be completely simplified.
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Basics of Graphing: Give a graph with appropriate scales and labels. The line should have the correct intercepts and general behavior.
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Problem structure: Solution should be organized and easy to follow, with clear, complete steps that follow progressively from one to the next.
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Mathematical Communication: Solutions should be presented in a clear and logical manner, with appropriate use of mathematical notation and terminology. Equations should have at most one equal sign and expressions on both sides.
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Set up and solve compound interest problems. Expressions should be completely simplified.
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Set up and solve continuous compound interest problems.
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Set up exponential growth problems and find the growth rate. Answers should be simplified fractions, integers, or in terms of natural logarithms.
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Use the growth rate and the model to make predictions. Answers should be simplified fractions, integers, or in terms of natural logarithms and have correct units.
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Solve exponential growth problems for specific time periods. Leave answers as simplified fractions, integers, or in terms of natural logarithms and have correct units.
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1.

A new employee wants to invest $2000 into an account that pays 6% annual interest compounded quarterly.

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(a)

Set up the simplified expression that would represent the amount of money in the account at time, $t\text{,}$ years.

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Hint.

Use the compound interest formula.

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Answer.

$A(t)=2000\left(\frac{203}{200}\right)^{4t}$

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Solution.

The compound interest formula is:

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$A(t)=P\left(1+\frac{r}{n}\right)^{nt}$

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Where:

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$P=2000\text{,}$ $r=0.06\text{,}$ $n=4\text{,}$ and $t$ is the time in years.

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Substituting these values gives:

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$A(t)=2000\left(1+\frac{0.06}{4}\right)^{4t}\text{.}$ Simplifying the base gives:

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$A(t)=2000\left(\frac{203}{200}\right)^{4t}$

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(b)

How long would it take for the account to reach $5000? (Leave answer as a simplified expression in terms of natural logarithms.)

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Hint.

Set $A(t)=5000$ and solve for $t\text{.}$

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Answer.

$t=\frac{\ln\left(\frac{5}{2}\right)}{4\ln\left(\frac{203}{200}\right)}$

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Solution.

Setting $A(t)=5000$ gives:

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$5000=2000\left(\frac{203}{200}\right)^{4t}$

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Dividing both sides by 2000:

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$\frac{5}{2}=\left(\frac{203}{200}\right)^{4t}$

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Taking the natural logarithm of both sides:

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$\ln\left(\frac{5}{2}\right)=4t\ln\left(\frac{203}{200}\right)$

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Solving for $t\text{:}$

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$t=\frac{\ln\left(\frac{5}{2}\right)}{4\ln\left(\frac{203}{200}\right)}$

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2.

A certain animal population is modeled by the function $P(t)=\frac{2000}{1+2^{-.05t}}\text{,}$ where $P(t)$ is the population at time $t$ months.

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(a)

What is the initial population for this animal?

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Hint.

Substitute $t=0$ into the function to get the initial population.

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Answer.

$P(0)=1000$ animals

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Solution.

Substituting $t=0$ into the function gives:

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$P(0)=\frac{2000}{1+2^{0}}=\frac{2000}{1+1}=\frac{2000}{2}=1000$

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(b)

After 10 months, what is the population? (Leave answer as a simplified expression.)

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Hint.

Substitute $t=10$ into the function.

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Answer.

$P(10)=2000\sqrt{2}-2000$ animals

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Solution.

Substituting $t=10$ into the function gives:

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\begin{align*} P(10) &=\frac{2000}{1+2^{-0.5}}\\ &=\frac{2000}{1+\frac{1}{2^{0.5}}}\\ &=\frac{2000}{1+\frac{1}{\sqrt{2}}}\\ &=\frac{2000(\sqrt{2}-1)}{\frac{2-1}}\\ &=2000(\sqrt{2}-1) \end{align*}

animals

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