Applications of Exponential Growth

Interest rates

If a bank account pays 5% yearly interest, that means if you have \$$x$ in the account, then after a year the bank will add 5% of \$$x$ to the account. Since 5% of \$$x$ is the same as \$$0.05x$, if you start with \$$x$, a year later you will have $\$x+\$0.05x=\$1.05x$. So every year, the amount of money in the account is multiplied by 1.05.

After three years, the amount of money in the account has been multiplied by $1.05 ⋅ 1.05 ⋅ 1.05=1.05^3$. What has the amount of money in the account been multiplied by after five years?
What has it been multiplied by after $n$ years?
If you put \$1000 in a bank account that pays 5% yearly interest, then after 4 years you will have $\$1000 ⋅ 1.05^4=\$1215.51$ (to the nearest cent). How much money will you have after 7 years?
If a bank account pays 3% yearly interest, how much will the amount of money in it be multiplied by after 1 year?
If a bank account pays 3% yearly interest, how much will the amount of money in it be multiplied by after 2 years?
If a bank account pays 3% yearly interest, how much will the amount of money in it be multiplied by after $n$ years?
If you put \$1000 in a bank account that pays 3% yearly interest, how much money will you have after 7 years?

In general:

If a bank account pays $r$% yearly interest, then after $n$ years the amount of money in your account will be multiplied by $$(1+r/100)^n$$.

If a credit card charges 2% monthly interest, that means if you owe \$$x$, then after a month the credit card company will charge you 2% of \$$x$. You will then owe a total of $\$x+\$0.02x=\$1.02x$. So each month, the amount of money you owe is multiplied by 1.02.

How much has the amount you owe been multiplied by after three months?
How much has the amount you owe been multiplied by after $m$ months?
If you owe \$1000 on the credit card, how much money will you owe in 1 year (12 months)?

If your debt is being multiplied by $$1+r/100$$ every year, you are being charged $r$% yearly interest. For example, if it was being multiplied by $1.15$ every year, then that would mean you were being charged 15% yearly interest.

How much is your debt being multiplied by each year? Round to two decimal places.
If your credit card company charges 2% monthly interest, what is the approximate yearly interest that they charge? %

Musical pitches

In music, the pitch of a note (how high or low it is) is determined by its frequency (number of vibrations per second). Two pitches are an octave apart if the higher one has a frequency which is twice that of the lower one. For example, a pitch with a frequency of 200 vibrations per second is an octave above a pitch with a frequency of 100 vibrations per second.

The pitch “A440,” which is the standard reference pitch, has a frequency of 440 vibrations per second. What is the frequency of the pitch an octave above A440? vibrations per second

If we want to find the frequency of the pitch two octaves below A440, we should multiply 440 by $$1/2$$ twice, to get a frequency of $$440 ⋅ 1/2 ⋅ 1/2 = 440 ⋅ (1/2)^2 = 110$$ vibrations per second.

What is the frequency of the pitch three octaves below A440?

vibrations per second

It’s standard to divide an octave into twelve equal semitones. Because all semitones are equal, and going up by twelve semitones multiplies a frequency by 2, going up by one semitone multiplies a frequency by $2^{1∕12}$.

What is the frequency of the pitch one semitone above A440? Round to the nearest whole number. vibrations per second

When rounded to the nearest whole number, the pitch three semitones above A440 has a frequency of

$440 ⋅ (2^{1∕12})^3=440 ⋅ 2^{3∕12} = 440 ⋅ 2^{1∕4} ≈ 523$ vibrations per second

When rounded to the nearest whole number, what is the frequency of the pitch six semitones above A440?

vibrations per second
If you want to go up seven semitones, what do you need to multiply a pitch’s frequency by (to four decimal places)?

Musicians often find that pitches whose frequencies are approximately in small whole-number ratios to each other, like $$2/1$$, $$4/3$$, $$3/2$$, etc., sound good when played together. These pairs of pitches are called consonant. Because going up by seven semitones leads to multiplication by a number which is very close to $$3/2$$, two notes that are seven semitones apart are consonant. But there is no small fraction which is close to the frequency multiplier for six semitones, so two notes that are six semitones apart are not consonant.

Spread of a disease (Ebola)

$x$$y$Date
$0$$107$June 30
$2$$115$July 2
$6$$131$July 6
$8$$142$July 8
$12$$172$July 12
$14$$174$July 14
$17$$196$July 17
$20$$224$July 20

Let’s look at the spread of the Ebola virus in Liberia during the 2014 outbreak, and see whether it grew at a linear, quadratic, or exponential rate.

Look at the table to the left. The numbers in the $x$ column tell us how many days elapsed since June 30, 2014. The numbers in the $y$ column tell us the total number of Ebola cases in Liberia on that date.

On June 30 there were 107 known cases of Ebola in Liberia. How many known cases were there on July 2?
How many known cases were there on July 6?
Over the time shown in the table, was the number of cases of Ebola increasing or decreasing?

The equation $y=5.7x+101$ is the linear equation that comes closest to passing through the data points. It is graphed on the grid to the left.

To the right of the graph you can see that, when $x=25$, $y = 243.5 ± ~5.234$. This means that the equation predicts that, after 25 days, there would be 243.5 diagnoses. The number 5.234 is called the root-mean-square error of the graph. This number tells you how well the equation represents the data (a smaller number means the equation is a better “fit” to the data).

Click on the graph, and drag the vertical bar to the left or right. On the right hand side of the graph you will see the graph’s $y$ value for each $x$ value you choose.

Move the vertical bar so that $x=30$. What value does $y$ have when $x=30$?
What value does $y$ have when $x=40$?
If we use this equation to approximate the number of Ebola infections, how many infections does it predict 50 days after the beginning of the data (that is, on August 19)?

Now let’s see how well a quadratic equation can fit the data. The equation $y=a(x-h)^2+k$ is graphed on the grid to the left, with sliders for $a$, $h$, and $k$.

Use the sliders for $a$, $h$, and $k$ to find a better fit to the data by watching the graph and the root-mean-square error to the right of the grid. As a starting point, set $h$ to $-16$ and $k$ to 79. Then, slide $a$ to make the error as small as you can (it should end up below 4.0).

What quadratic equation did you find?
What is the root-mean-square error for the equation you found?

If you look back at linearEbolaQn, you will see that the root-mean-square error for the best linear equation was 5.234.

Which equation, the linear equation from linearEbolaQn or the quadratic equation you found in this question, is a better fit to the data?
As in linearEbolaQn, use the mouse to click and drag the vertical bar on the graph. What does the quadratic equation you found predict for the number of Ebola cases on August 19, 2014 (that is, when $x = 50$)?

Now let’s see how well an exponential equation fits the data. The equation $y=m ⋅ a^x$ is graphed on the grid to the left, with sliders for $a$ and $m$.

Use the sliders for $a$ and $m$ to find a better fit to the data by watching the graph and the root-mean-square error to the right of the grid. Make the error as small as you can (it should end up below 4.0). What exponential equation did you find? (Try switching back and forth between the sliders. First make the error as small as possible by using only the $a$ slider. Then make it as small as possible at that $a$ value by using only the $m$ slider. Then make it as small as possible at that $m$ value by using only the $a$ slider, and so on until it’s small enough.)
What is the root-mean-square error for the equation you found?
What does this equation predict for the total number of cases of Ebola on August 19, 2014 (when $x=50$)?
Compare the root-mean-square error for the three equations you’ve looked at in the last three questions. Two of the three errors should be significantly smaller than the third one. Of the linear, the quadratic, and the exponential equation, which two are better fits for the data?

As you saw in the previous question, the quadratic and exponential equations seemed to do a better job of fitting the data than the linear equation. However, it is also important to look at how well an equation does at predicting what happens after the end of our initial data.

The actual number of cases of Ebola in Liberia on August 19 was 1027. Look back at your quadratic and exponential predictions from the previous questions. Did the quadratic or the exponential equation come closest to predicting the actual number of cases?
On March 5, 2015, it was reported that there had been no new cases of Ebola in Liberia in the previous week. Did any of the three equations predict that the number of cases would stop increasing?