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Algebra II
Exponential Growth and Decay
2015-11-19 www.njctl.org
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Simple Annual Interest Compound Interest Half-Lives & Decay Applications Population Growth PARCC Sample Questions Standards The Constant, e
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Simple Annual Interest
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One important reason to invest your money is the opportunity to earn interest; which means your bank pays you money for keeping it in
The money you earn depends on the percentage interest you are paid per time period and how long your money is in the account. There are a few different ways interest can be calculated, but simple interest is earned based on the initial investment amount only.
Simple Interest Slide 6 / 128
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In general, this becomes Where A is the accrued amount P is the principal (initial investment) r is the interest rate for that time period t is the time invested
Simple Interest Slide 9 / 128 Simple Interest
Continuing with our example... If you are paid 10% simple interest per year on your initial investment of $1000, what would be your account balance after 3 years?
Slide 10 / 128 Simple Interest
With simple interest, your interest is always calculated based on your initial investment, or starting principal. You can see that the $100 remains the same each year because the initial investment was $1000. Year Account Balance Interest $1000 1 $1100 $100 2 $1200 $100 3 $1300 $100 4 $1400 $100
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1 Which equation describes your ending bank balance if $1000 earns 5% simple annual interest for 7 years? A B C D E None of these
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2 Which equation describes your ending bank balance if $500 earns 6% simple annual interest for 3 years? A B C D E None of these
SLIDE 3
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3 What will be your bank balance if you put $600 in your account and earn 5% simple annual interest for seven years?
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4 What will be your bank balance if you put $1800 in your account and earn 4% simple annual interest for six years?
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5 What will be your bank balance if you put $3000 in your account and earn 2% simple annual interest for ten years?
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6 If you are earning 7% simple annual interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
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7 If you are earning 10% simple annual interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
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8 If you are earning 2% interest and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
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Compound Interest
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Slide 20 / 128 Compound Interest
Compound interest can be thought of as "making interest on interest." Every time the interest is calculated, the current account balance is used to calculate the new interest. This means you are earning slightly more each time period (assuming the other factors are constant) compared to simple interest.
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Recalling our example from the first section, if you are paid 10% simple interest per year on your balance of $1000, you would be paid $100 at the end of one year so your balance at the end of one year is $1100. With compound interest, the following years you will earn interest not
- nly on your original $1000, but also the interest you've earned in
prior years. This is called the compounding effect of interest. In the real world, it is better to be earning compounding interest than to be paying it...it grows very fast. That's why saving and investing early is so important. At the same time, this is why it can be hard to get out of debt, when you're on the wrong side of compounding interest.
Compound Interest Slide 22 / 128
Year Balance Interest $1000 $100 1 $1100 $110 2 $1210 $121 3 $1331 $133.1 4 $1464.1 $146.41 5 $1610.51
Compound Interest
Earning 10% compound interest, yield the table below. Notice, the interest is calculated based on the previous year's ending balance.
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Why does the amount of interest earned increase each year? Instead of total interest of $500 (with simple interest), you earn $610.51. Why?
Compound Interest
Year Balance Interest $1000 $100 1 $1100 $110 2 $1210 $121 3 $1331 $133.1 4 $1464.1 $146.41 5 $1610.51
Math Practice
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Algebraically, After two years, the amount you earn would be given by But we can rewrite this expression to yield: What do you think your account balance will be after three years?
Compound Interest
SLIDE 5
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Therefore, in general, your account balance with compound interest will be given by where A(t) is the amount of money after t time periods P is the principal, or initial investment t is the number of time periods (usually years) r is the interest rate per time period
Compound Interest Slide 26 / 128 Compound Interest
Practice: Calculate the total account balance after investing $750 at 5% interested compounded yearly for 8 years.
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With annual interest, you receive your interest at the end of the time period, in this case the year. But, it's also possible for interest to compound within the year. For instance, your interest rate could be compounded quarterly. In this case, the interest is paid four time each year. The number of times per year that interest is compounded is called n. So, in this case, n = 4.
Compound Interest Slide 28 / 128
If n = 4, that means that we calculate and pay interest four times. It also means that only 1/4 of a year will have passed between each interest calculation. So, we have to divide the annual interest rate by 4 to get the interest rate for one calendar quarter: 10% divided by 4 = 2.5% Then we calculate the interest 4 times. The power of 4 reflects that the interest is calculated four times a year, each time at the annual rate divided by 4.
Quarterly Compounding Slide 29 / 128
So, even though the annual interest rate is the same: 10% In this case, you earn an extra $3.81 by quarterly compounding as compared to annual interest. You end with $1103.81 rather than $1100.00
Quarterly Compounding Slide 30 / 128
In general, the result of compounding more frequently is given by the formula: where A is the total account balance P is the principal, or starting balance r is the annual interest rate t is the number of years n is the number of times per year that the interest is compounded
Compounding
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What if we compounded weekly? What would the formula look like for that? Discuss and write a formula for that case. Then, determine your bank balance after one year, starting with $1000 and compounding weekly with 10% interest.
Weekly Compounding Slide 32 / 128
When we compounded 4 times, we gained $3.08 more than if we had used simple interest Compounding 52 times earns a bit more: $5.06 is $1.98 more than the $3.08 we earned by compounding 4 times. Let's see what happens if we keep increasing our compounding
Weekly Compounding Slide 33 / 128
Fill in this chart for compounding: Daily (365.25 times) Each second (31,557,600 times)
Compound Interest
Interest Balance Annual $100 $1100 Quarterly $103.81 $1103.81 Weekly $5.06 $1105.06 Daily Every second
Math Practice
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Compounding Interest Balance Annually $100 $1100 Quarterly $103.81 $1103.81 Weekly $5.06 $1105.06 Daily Every second
Compound Interest
Answer
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9 Which equation describes your bank balance if $5250 earns 4% annual interest, compounded annually for 9 years? A B C D E None of these
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10 Which equation describes your bank balance if $1000 earns 6% annual interest, compounded quarterly, for 7 years? A B C D E None of these
SLIDE 7
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11 Which equation describes your ending bank balance if $500 earns 9% annual interest, compounded monthly, for 3 years? A B C D E None of these
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12 What will your bank balance be if you put $600 in your account and earn 5% interest, compounded weekly, for seven years?
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13 What will be your bank balance if you put $1800 in your account and earn 4% interest, compounded daily, for six years?
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14 What will be your bank balance if you put $3000 in your account and earn 2% interest, compounded weekly, for ten years?
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15 If you are earning 7% interest, compounded daily, and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
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16 If you are earning 10% interest, compounded weekly, and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
SLIDE 8
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17 How much was originally invested if you have $63,710.56 in an account generating 4% interest (compounded monthly) over 15 years?
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18 What interest rate is needed to double your money if it's invested for 8 years compounded quarterly?
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Let's look at this data from a slightly different perspective: Compounding Times Per Year Interest Extra Interest Earned 1 $100.00 4 $103.81 $3.81 52 $105.06 $1.25 365.25 $105.16 $0.10 31,557,600 $105.17 $0.01 What do you notice about the amount of interest gained as the amount of compounding increases?
Compound Interest Slide 46 / 128
The amount of interest earned is approaching $105.17 per year as the amount of compounding gets very large. Mathematicians wondered if there was a way to predict the limit as the amount of compounding approached infinity (if compounding was done continuously.) The next section will address this intriguing question! Compound Times Per Year Interest Extra Interest Earned 1 $100.00 4 $103.81 $3.81 52 $105.06 $1.25 365.25 $105.16 $0.10 31,557,600 $105.17 $0.01
Compound Interest Slide 47 / 128 Recap:
Discuss: What is the difference between simple and compound interest?
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The Constant, e
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Let's focus in on a portion of our formula Specifically looking at just one year, let t = 1 Now using the special case of r = 1 Does this approach a number as n gets very large?
The Constant, e Slide 50 / 128
n 1 10 100 1000 10,000 100,000 Use your calculator to complete this table.
The Constant, e Slide 51 / 128
n 1 2 10 2.5937 100 2.7048 1000 2.7169 10,000 2.7181 100,000 2.7182 Infinity Mathematicians proved that as n approaches infinity, approaches the number they were to name e.
The Constant, e Slide 52 / 128
n 1 2 10 2.5937 100 2.7048 1000 2.7169 10,000 2.7181 100,000 2.7182 Infinity 2.7182... e is an irrational number: its decimals go on forever and never end, repeat or form a pattern. It's first few digits are 2.7182.
The Constant, e Slide 53 / 128
2.718281828459045235360287471352662497757247093699959574966967627 7240766303535475945713821785251664274274663919320030599218174135 9662904357290033429526059563073813232862794349076323382988075319 5251019011573834187930702154089149934884167509244761460668082264 8001684774118537423454424371075390777449920695517027618386062613 3138458300075204493382656029760673711320070932870912744374704723 0696977209310141692836819025515108657463772111252389784425056953 6967707854499699679468644549059879316368892300987931277361782154 2499922957635148220826989519366803318252886939849646510582093923 9829488793320362509443117301238197068416140397019837679320683282 3764648042953118023287825098194558153017567173613320698112509961 8188159304169035159888851934580727386673858942287922849989208680 5825749279610484198444363463244968487560233624827041978623209002 1609902353043699418491463140934317381436405462531520961836908887 0701676839642437814059271456354906130310720851038375051011574770 41718986106873969655212671546889570350354
Here are the first 1000 digits of e, but e has been calculated to millions of digits.
The Constant, e Slide 54 / 128
Your calculator actually has a key for e since it has many uses! e shows up in a wide range of math, science, business, economics, medicine, etc. It was first used, but not identified specifically, by John Napier in about 1618.It was derived by Jacob Bernouli and then used by Leibniz and Huygens in the late 1600s.The designation of the letter e for this number was made by Euler in the early 1700s. It was first computed from solving the problem we've been exploring...compound interest. Since then, it has been computed many different ways.
The Constant, e
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The letter e allows us to simplify the continuous compounding of interest (n going to infinity) by replacing this formula: with this formula where A is the amount after t time periods P is the principal, or starting amount r is the growth rate per time period t is the number of time periods
Continuous Compounding Slide 56 / 128
Let's check this by substituting into the formula for our example of $1000 earning 10% interest compounded continuously.
Continuous Compounding
The account balance after one year of being compounded continuously is $1,105.17.
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We can now add a row for continuous compounding by using e in
Compound Times Per Year Interest Extra Interest Earned 1 $100.00 4 $103.81 $3.81 52 $105.06 $1.25 365.25 $105.16 $0.10 31,557,600 $105.17 $0.01 Continuous $105.17 < $0.01
Continuous Compounding Slide 58 / 128
19 Which equation describes your bank balance if $1000 earns 6% annual interest, compounded continuously, for 7 years? A B C D E None of these
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20 Which equation describes your ending bank balance if $500 earns 9% annual interest, compounded continuously for 3 years? A B C D E None of these
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21 What will your bank balance be if you put $600 in your account and earn 5% annual interest, compounded continuously, for seven years?
SLIDE 11
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22 What will be your bank balance if you put $1800 in your account and earn 4% annual interest, compounded continuously, for six years?
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23 What will be your bank balance if you put $3000 in your account and earn 2% annual interest, compounded continuously, for ten years?
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24 If you are earning 7% annual interest, compounded continuously, and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
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25 If you are earning 10% annual interest, compounded continuously, and your goal is to have $3000 in your account after six years, how much will you have to initially deposit?
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Population Growth
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Slide 66 / 128 Population Growth
Sometimes, you will be asked questions that are related to the interest formulas, but presented slightly different. Here's an example: A town's population is growing at a rate of 3% per year. If its initial population is 1.2 million people, what will its population be in 5 years? Which formula would we use for this problem?
SLIDE 12 Slide 67 / 128 Population Growth
A town's population is growing at a rate of 3% per year. If its initial population is 1.2 million people, what will its population be in 5 years? The familiar formula we can use to answer this question is where A is the population at some time, t P is the initial population r is the growth rate (written as a decimal) t is the time in years after the initial population was measured
Slide 68 / 128 Population Growth
A town's population is growing at a rate of 3% per year. If its initial population is 1.2 million people, what will its population be in 5 years, in millions? Now, we can substitute our values into the equation and solve. The town's population will be 1.391 million people Populations often grow based on this exponential model.
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26 A town's population is growing at a rate of 4.5% per
- year. If its initial population is 2.3 million people, what
will be its population in 3 years?
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27 A town's population is growing at a rate of 5.3% per
- year. If its initial population is 1.4 million people, what
will be its population in 7 years?
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28 A town's population is growing at a rate of 6.25% per
- year. If its initial population is 2.15 million people, what
will be its population in 11 years?
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29 A town's population is growing at a rate of 7.75% per
- year. If its initial population is 2.8 million people, what
will be its population in 13 years?
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In other scenarios, you will be provided with an equation that contains some of the information already provided, such as where A(t) is the population at some time, t P is the initial population, t is the time in years after the initial population was measured 1.5 can be translated into (1 + 0.5), which means that the growth rate was 0.5, or 50%. Let's use this model for the next questions.
Population Models Slide 74 / 128
If the initial population is 4 million, what will the population be after
Example Slide 75 / 128
If the initial population is 4 million, what will the population be after six years?
Example Slide 76 / 128
If the initial population is 4 million, what will the population be after five months? You might think that you could just recall that after one year the population is 6 million and it started at 4 million, so after half a year it'd be 4 million + of 2 million = 5.83 million But, that would only be true for a linear model...and this is
- exponential. So this is not correct.
Discuss possible solutions.
Example Slide 77 / 128
We convert the formula to provide monthly results by raising the rate to the power of 12t, rather than t, so that when we put in the number of years as t, it converts that to 12 t months. In order not to change the amount after one year, I have to take the 12th root of the rate, so that the equation is still true NOTE: The properties of exponents shows that Substitute t = , since we are looking for of a year
Example Slide 78 / 128
Substitute t = , since we are looking for of a year
Example
SLIDE 14
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where the value of t is measured in years. To convert and be able to substitute months, we must divide t by 12.
Example
Instead of changing our exponential base to solve the problem, we could also change the form of the exponent. In our original problem, our function was
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If the initial population is 4 million, what will the population be after 15 months?
Example Slide 82 / 128
30 Using the below model, calculate the population, in millions, after one year if the initial population is 5.8 million.
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31 Using the below model, calculate the population, in millions, after five years, if the initial population is 5.8 million.
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32 Which equation would describe the monthly population consistent with the given annual model. A B C D E None of the above
SLIDE 15
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33 Which equation would describe the quarterly population consistent with the given annual model. A B C D E None of the above
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34 Using this annual model for population growth, calculate the population, in millions, after three months.
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35 Using this annual model for population growth, calculate the population, in millions, after 16 months.
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36 Using this annual model for population growth, calculate the population, in millions, after 19 months.
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Half-Lives and Decay Applications
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There are nuclear reactions that occur which result in a nucleus having fewer protons or neutrons. If the number of protons changes, the result is a new element. If only the number of neutrons changes, the element stays the same but it becomes a new isotope of that element. Spontaneous transformation (those that occur without outside forces) result in a reduction in the number of protons and/or neutrons...so it's called "nuclear decay."
Nuclear Decay
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It is impossible to predict if an individual nucleus will decay. But, given a large number of nuclei, the percentage which will decay in a fixed amount of time is predictable. The time it takes for half of the nuclei in a sample to decay is called the half-life of that isotope. After each half-life, half of the nuclei have decayed.
Nuclear Decay Slide 92 / 128
Since this rate of decay is so predictable, it can be used to date how long a substance was created. This is done by measuring the ratio of the amount of the original nuclei to the decayed nuclei. From that, the number of half-lives which have occurred is calculated. Since half-lives are known and predictable, the age of the object can be determined.
Nuclear Decay Slide 93 / 128
When plants grow they take in Carbon from the atmosphere. That Carbon decays from Carbon-14 to Nitrogen-14 over many years. By looking at the ration of those elements, how old the living material is can be determined. The half-life of that reaction is 5,730 years.
Carbon Dating Slide 94 / 128
Each half life results in a loss of half the original substance.
Half Lives
https://en.wikipedia.org/wiki/Half-life
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The amount of remaining substance over time is shown here.
Half Lives Slide 96 / 128
SLIDE 17 Slide 97 / 128
can be an amount of mass, m, or the number of nuclei, N
Half Life Formula Slide 98 / 128
From our previous formula, n was the number of half-lives which have elapsed. It can be calculated on its own using the formula where t is the time elapsed , tau, is the half-life of the material
Half Life Formula
Combining our formulas yields
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After 250 days, how many grams will remain of a 100 gram sample if the for that substance is 75 days?
Example Slide 100 / 128
Teachers: Use the questions located in the pull tab for the next 6 slides.
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37 If 1/4 of the initial substance remains, how many half- lives have elapsed?
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38 If the half-life of that substance is 22 days, and only 1/4
- f it remains, how many days have elapsed?
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39 If 1/16 of an initial substance remains, how many half- lives have elapsed?
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40 If the half-life of that substance is 80 days, and only 1/16
- f it remains, how many days have elapsed?
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41 If 12.5% of an initial amount of Carbon 14 remains, how many years ago did the plant die? ( , for Carbon-14 is 5730 years.)
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42 A certain radioactive material has a half-life of 20 years. If 100g were present to start, how much will remain in 7 years?
Slide 107 / 128 Additional Applications
Sometimes, you will be asked questions that are related to the interest formulas, but are slightly different. Here's an example: A new car depreciates in value at a rate of 8% per year. If you purchase it for $31,000, how much is it worth after 10 years? Which formula would we use for this problem?
Slide 108 / 128 Additional Applications
A new car depreciates in value at a rate of 8% per year. If you purchase it for $31,000, how much is it worth after 10 years? The formula that we can use to answer this question is How will we write the rate?
SLIDE 19 Slide 109 / 128 Additional Applications
A new car depreciates in value at a rate of 8% per year. If you purchase it for $31,000, how much is it worth after 10 years? Since the car's value is falling at a rate is 8%, it is given a negative value, so our equation would become
Slide 110 / 128 Additional Applications
A new car depreciates in value at a rate of 8% per year. If you purchase it for $31,000, how much will it be worth after 10 years? Now, we can calculate our answer Therefore, the car will be worth $13,466.04 in 10 years.
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43 A new truck depreciates in value at a rate of 9% per year. If you purchase one for $27,150, how much is it worth after 5 years?
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44 A new SUV depreciates in value at a rate of 9.5% per
- year. If you purchase one for $33,725, how much will it
be worth after 7 years?
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45 A new sofa depreciates in value at a rate of 22% per
- year. If you purchase one for $1,300, how much will it be
worth after 4 years?
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46 A new car depreciates in value at a rate of 8% per year. If a 5 year old car is worth $20,000, how much was it
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47 A new sofa depreciates in value at a rate of 22% per
- year. If a 5 year old sofa is worth $500, how much was it
- riginally worth?
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PARCC Sample Questions
The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions. Good Luck! Return to Table
Slide 117 / 128
48 During a 1 year period, a population of tropical insects grew according to the model P = P0(1.46)t, where P is the population, P0 is the initial population, and t is time in
- years. Which equation can be used to model the
approximate weekly growth rate? (Assume 52 weeks in a year.) A P = P0(1.0073)52t B P = P0(1.0088)52t C P = P0(1.0281)52t D P = P0(1.0371)52t
PARCC Released Question - PBA - Calculator Section
Question 2/13
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49 A scientist places 7.35 grams of radioactive element in a dish. The half-life of the element is 2 days. After d days, the number of grams of the element remaining in the dish is given by the function Which statement is true about the equation when it is rewritten without a fractional exponent? Select all that apply. A An approximately equivalent equation is B An approximately equivalent equation is C The base of the exponent in this form of the equation can be interpreted to mean that the element decays by 0.250 grams per day. D The base of the exponent in this form of the equation can be interpreted to mean that the element decays by 0.707 grams per day. E The base of the exponent in this form of the equation can be interpreted to mean that about 25% of the element remains from one day to the next day. F The base of the exponent in this form of the equation can be interpreted to mean that about 70.7% of the element remains from one day to the next day. PARCC Released Question - EOY - Calculator Section
Question 13/26
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PARCC Released Question - EOY Calculator Section An investor deposits g dollars into an account at the beginning of each year for n years. The account earns an annual interest rate of r, expressed as a decimal. The amount of money S, in dollars, in the account can be determined by the formula Part A Suppose the investor deposits $500 for 10 years into an account that earns an annual interest rate of 5%. If no additional deposits or withdrawals are made, what will be the balance in the account at the end of 10 years?
- A. $6,003.05
- B. $6,015.06
- C. $6,288.95
- D. $6,301.52
Part B Suppose the investor wanted the balance in the account to be at least $12,000 at the end of 10 years. At an annual interest rate of 5%, the amount of the yearly deposit should be at least $_______. (answer to the nearest cent)
Question 19/26
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50 Part A - Suppose the investor deposits $500 a year for 10 years into an account that earns an annual interest rate
- f 5%. If no additional deposits or withdrawals are made,
what will be the balance in the account at the end of 10 years? A $6,003.05 B $6,015.06 C $6,288.95 D $6,301.52
PARCC Released Question - EOY Calculator Section
Question 19/26
SLIDE 21 Slide 121 / 128
51 PART B - Suppose the investor wanted the balance in the account to be at least $12,000 at the end of 10
- years. At an annual interest rate of 5%, the amount of
the yearly deposit should be at least $_______. (answer to the nearest cent)
PARCC Released Question - EOY Calculator Section
Question 19/26
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52 When approximating the age of an artifact that is less than 40,000 years old, the radioisotope carbon-14 can be used. Carbon-14 is an element with the property that every 5,730 years the mass of the element in a sample is reduced by half. The mass of the carbon-14 in an artifact can be modeled by an exponential function, m of its age, x. Part A Let A represent the original mass of carbon-14. Which function is an appropriate model?
A B C D
PARCC Released Question - EOY - Calculator Section
Question 26/26
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53 When approximating the age of an artifact that is less than 40,000 years old, the radioisotope carbon-14 can be used. Carbon-14 is an element with the property that every 5,730 years the mass of the element in a sample is reduced by half. The mass of the carbon-14 in an artifact can be modeled by an exponential function, m of its age, x. Part B Based on the situation, which interval represents the domain of the function m? A B C 0 ≤ x ≤ 5,730 D 0 ≤ x ≤ 40,000 PARCC Released Question - EOY - Calculator Section
Question 26/26
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54 When approximating the age of an artifact that is less
than 40,000 years old, the radioisotope carbon-14 can be used. Carbon-14 is an element with the property that every 5,730 years the mass of the element in a sample is reduced by half. The mass of the carbon-14 in an artifact can be modeled by an exponential function, m of its age, x. Part C Which statements describe the graph of m in the coordinate plane? Select all that apply. A The function m is a linear function. B The function m is a nonlinear function. C The function m is an increasing function. D The function m is a decreasing function. E The function m is a periodic function.
PARCC Released Question - EOY - Calculator Section
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55 When approximating the age of an artifact that is less
than 40,000 years old, the radioisotope carbon-14 can be used. Carbon-14 is an element with the property that every 5,730 years the mass of the element in a sample is reduced by half. The mass of the carbon-14 in an artifact can be modeled by an exponential function, m of its age, x. Part D At what age would the mass of carbon-14 in an artifact be one-fourth the original amount? A 1,432.5 years old B 2,865 years old C 11,460 years old D 22,920 years old
PARCC Released Question - EOY - Calculator Section
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Standards
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SLIDE 22 Slide 127 / 128
Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of
MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
Slide 128 / 128
