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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.
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.
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.
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.
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?
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 $2^{1∕12}$.
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?
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.
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.
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 after the $~$ 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.
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).
If you look back at linearEbolaQn, you will see that the root-mean-square error for the best linear equation was 5.234.
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 $m$.
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.