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Exponents Homework -- Real Life Exponential Curves

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample problems designed to develop real life applications of exponent functions.

Exercise 1

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.

Table 1
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.

Exercise 2

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.

Table 2
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?

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