5.7 Homework: “real life” exponential curves

First “radioactive decay” case

• a After $n$ 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.

Second “radioactive decay” case

• 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 functio n, 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?

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:

$M=A{\left(1+\frac{i}{n}\right)}^{\text{nt}}$

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

$1,000 ${\left(1+\frac{0\text{.}\text{10}}{2}\right)}^{2\times \text{30}}$ =$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?