Applications

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Application of Exponentials

Compound Interest

Example:

Suppose that you put $2,000 into a bank account that pays 6% interest compounded monthly. How much will you have in 5 years?

Solution

After the first month, the new balance will be

A = P(1 + rt) = 2,000(1 + (.06)(1/12))

the next month's balance is

2,000(1 + .06/12)(1 + .06/12) = 2,000(1 + .06/12)² = 2,000(1.005)²

The third month, the balance will be

2,000(1.005)²(1.005) = 2,000(1.005)³

After t months, the balance will be

2,000(1.005)^t

Five years is 60 months so the final balance will be

2,000(1.005)⁶⁰ = $2697.70

In general for an account that initially has P dollars in it and is left for t years in an account that pays interest at a rate of r and compounds m times per year we have

A = P(1 + r/m)^mt

For continuous compounding such as inflation, the formula is

A = Pe^rt

Exercise:

If health care costs $300 per month for the average family, how much will health care cost in the year 1050 if the inflation rate is 8% per year?

Radioactive Dating

If today there is P_o grams of a certain radioactive isotope, then after t years there will be

P = P_oe^rt

Example

You find a skull in a nearby Native American ancient burial site and with the help of a spectrometer, discover that the skull contains 9% of the C-14 found in a modern skull. Assuming that the half life of C-14 is 5730 years, how old is the skull?

First we use the fact that after 5730 years, there is half remaining so that

1/2P_o = P_oe^rt

0.5 = e^r5730

ln 0.5 = r(5730)

                  ln 0.5
        r =                   = -0.00012
                  5730

Since today there is .09P_o we have

0.09Po = Poe^-.00012t

0.09 = e^-.00012t

ln0.09 = -0.00012t

t = (ln.09)/-.00012 = 20,000 years old.

Exercises

Suppose that there is a fruit fly infestation in the central valley. Being an environmentalist, you propose a plan to spread 50,000 infertile fruit flies in the area to control the situation. Presently, you have in your laboratory 1,000 fruit flies. In 1 week they will reproduce to a population of 3,000 fruit flies. The farmers want to know when you will be ready to drop your infertile fruit flies. What should you tell them?
During the first month of the black plague there were 2,000 people infected. The following month there were 3,800 people infected. How long did it take until there were 500,000 people infected?

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