Exponential Decay?

Question

Inflation is causing things to cost roughly 2% more per year.
a. A bag of milk costs $3.75 now. Estimate its cost in five years.
b. A movie ticket costs $8.50 now. If inflation continues at 2% per year, when will the ticket cost $10.00?

Answer 1

This wouldn't be Exponential Decay, this would be Exponential Growth, if it were Exponential Decay, b. would not make any since.

A = Ao * e^(rt)

a.)
A = 3.75 * e^(.02 * 5)
A = 3.75 * e^(.1)
A = $4.14

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b.)
10 = 8.5 * e^(.02t)
(10/8.5) = e^(.02t)
(20/17) = e^(.02t)
ln(20/17) = .02t
t = (ln(20/17))/(.02)
t = about 8 years or 8.125946 years

Answer 2

This is actually exponential growth, as the amounts are increasing. What you need to to is change 2% to a decimal (0.02), add this to 100% as a decimal (1.00) to get the base of your exponent (1.02).

The equation takes the form
(final amount) = (initial amount)(percent as decimal)^t where t = number of years

So here that's F = 3.75(1.02)^5

you can calculate that directly.

The other one is tougher because the exponent t is unknown.

10.00 = 8.50(1.02)^t

you need to divide both sides by 8.50, then taKe the log of both sides, which changes the t from an exponent to a factor:

log(10.00/8.50) = t log 1.02
Now solve for t by dividing