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Homework: “Real life” exponential curves

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

Summary: This module provides practice problems designed to explore realistic applications of exponents.

Radioactive substances decay according to a “half-life.” The half-life is the period of time that it takes for half the substance to decay. For instance, if the half-life is 20 minutes, then every 20 minutes, half the remaining substance decays.

As you can see, this is the sort of exponential curve that goes down instead of up: at each step (or half-life) the total amount divides by 2; or, to put it another way, multiplies by ½.

Exercise 1

First “Radioactive Decay” Case

You have 1 gram of a substance with a half-life of 1 minute. Fill in the following table.

Table 1
Time Substance remaining
0 1 gram
1 minute ½ gram
2 minutes  
3 minutes  
4 minutes  
5 minutes  
  • a. After nn minutes, how many grams are there? Give me an equation.
  • b. Use that equation to answer the question: after 5 minutes, how many grams of substance are there? Does your answer agree with what you put under “5 minutes” above? (If not, something’s wrong somewhere—find it and fix it!)
  • c. How much substance will be left after 4½ minutes?
  • d. How much substance will be left after half an hour?
  • e. How long will it be before only one one-millionth of a gram remains?
  • f. Finally, on the attached graph paper, do a graph of this function, where the “minute” is on the x-axis and the “amount of stuff left” is on the y-axis (so you are graphing grams as a function of minutes). Obviously, your graph won’t get past the fifth or sixth minute or so, but try to get an idea for what the shape looks like.

Exercise 2

Second “Radioactive Decay” Case

Now, we’re going to do a more complicated example. Let’s say you start with 1000 grams of a substance, and its half-life is 20 minutes; that is, every 20 minutes, half the substance disappears. Fill in the following chart.

Table 2
Time Half-Lives Substance remaining
0 0 1000 grams
20 minutes 1 500 grams
40 minutes    
60 minutes    
80 minutes    
100 minutes    
  • a. After n half-lives, how many grams are there? Give me an equation.
  • b. After n half-lives, how many minutes have gone by? Give me an equation.
  • c. Now, let’s look at that equation the other way. After t minutes (for instance, after 60 minutes, or 80 minutes, etc), how many half-lives have gone by? Give me an equation.
  • d. Now we need to put it all together. After t minutes, how many grams are there? This equation should take you directly from the first column to the third: for instance, it should turn 0 into 1000, and 20 into 500. (*Note: you can build this as a composite function, starting from two of your previous answers!)
  • e. Test that equation to see if it gives you the same result you gave above after 100 minutes.
  • f. Once again, graph that do a graph on the graph paper. The x-axis should be minutes. The y-axis should be the total amount of substance. In the space below, answer the question: how is it like, and how is it unlike, the previous graph?
  • g. How much substance will be left after 70 minutes?
  • h. How much substance will be left after two hours? (*Not two minutes, two hours!)
  • i. How long will it be before only one gram of the original substance remains?

Exercise 3

Compound Interest

Finally, a bit more about compound interest

If you invest $A into a bank with i% interest compounded n times per year, after t years your bank account is worth an amount M given by:

M=A1+inntM=A1+innt size 12{ left (1+ { {i} over {n} } right ) rSup { size 8{ ital "nt"} } } {}

For instance, suppose you invest $1,000 in a bank that gives 10% interest, compounded “semi-annually” (twice a year). So AA, your initial investment, is $1,000. ii, the interest rate, is 10%, or 0.10. nn, the number of times compounded per year, is 2. So after 30 years, you would have:

$1,000 1+0.1022×301+0.1022×30 size 12{ left (1+ { {0 "." "10"} over {2} } right ) rSup { size 8{2 times "30"} } } {}=$18,679. (Not bad for a $1,000 investment!)

Now, suppose you invest $1.00 in a bank that gives 100% interest (nice bank!). How much do you have after one year if the interest is...

  • a. Compounded annually (once per year)?
  • b. Compounded quarterly (four times per year)?
  • c. Compounded daily?
  • d. Compounded every second?

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