5.6 Real life exponential curves

The famous king exponent story

This is a famous ancient story that I am not making up, except that I am changing some of the details.

A man did a great service for the king. The king offered to reward the man every day for a month. So the man said: “Your Majesty, on the first day, I want only a penny. On the second day, I want twice that: 2 pennies. On the third day, I want twice that much again: 4 pennies. On the fourth day, I want 8 pennies, and so on. On the thirtieth day, you will give me the last sum of money, and I will consider the debt paid off.”

The king thought he was getting a great deal…but was he? Before you do the math, take a guess: how much do you think the king will pay the man on the 30th day?

Now, let’s do the math. For each day, indicate how much money (in pennies) the king paid the man. Do this without a calculator, it’s good practice and should be quick.

Day 1: _1 penny_	Day 7: ________	Day 13: _______	Day 19: _______	Day 25: _______
Day 2: ___2____	Day 8: ________	Day 14: _______	Day 20: _______	Day 26: _______
Day 3: ___4____	Day 9: ________	Day 15: _______	Day 21: _______	Day 27: _______
Day 4: ___8____	Day 10: _______	Day 16: _______	Day 22: _______	Day 28: _______
Day 5: ________	Day 11: _______	Day 17: _______	Day 23: _______	Day 29: _______
Day 6: ________	Day 12: _______	Day 18: _______	Day 24: _______	Day 30: _______

How was your guess?

Now let’s get mathematical. On the nth day, how many pennies did the king give the man?

Use your calculator, and the formula you just wrote down, to answer the question: what did the king pay the man on the 30th day? _______ Does it match what you put under “Day 30” above? (If not, something’s wrong somewhere—find it and fix it!)

Finally, do a graph of this function, where the “day” is on the x-axis and the “pennies” is on the y axis (so you are graphing pennies as a function of day). Obviously, your graph won’t get past the fifth or sixth day or so, but try to get an idea for what the shape looks like.

Compound interest

Here is a slightly more realistic situation. Your bank pays 6% interest, compounded annually. That means that after the first year, they add 6% to your money. After the second year, they add another 6% to the new total…and so on.

You start with $1,000. Fill in the following table.

Year	The bank gives you this…	and you end up with this
0	0	$1000
1	$60	$1060
2	$63.60
3
4
5

Now, let’s start generalizing. Suppose at the end of one year, you have x dollars. How much does the bank give you that year?

And when you add that, how much do you have at the end of the next year? (Simplify as much as possible.)

So, now you know what is happening to your money each year. So after year n, how much money do you have? Give me an equation.

Test that equation to see if it gives you the same result you gave above for the end of year 5.

Once again, graph that. The x-axis should be year. The y-axis should be the total amount of money you end up with after each year.

How is this graph like, and how is it unlike, the previous graph?

If you withdraw all your money after ½ a year, how much money will the bank give you? (Use the equation you found above!)

If you withdraw all your money after 2½ years, how much money will the bank give you?

Suppose that, instead of starting with $1,000, I just tell you that you had $1,000 at year 0. How much money did you have five years before that (year –5)?

How many years will it take for your money to triple? That is to say, in what year will you have $3,000?