Question 1041884: The purchasing power of A dollars after t years of r% inflation is given by the model P = Ae^-rt. Assume the inflation rate is currently 3.8%. How long will it take for the purchasing power of $1.00 to be worth $0.76? Round the answer to the nearest hundredth.
I already got the answer, but it would be great if someone could explain how to solve it.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your formula is p = a * e^(-rt)
p is the future value
a is the present amount
e is the scientific constant e
r is the annual growth rate as a decimal
t is the number of years
3.8% is the annual growth rate as a percent.
divide it by 100 to get .038
that's the annual growth rate as a decimal.
p = .76
a = 1
formula becomes .76 = 1 * e^(-.038 * t)
simplify this to get .76 = e^(-.038 * t)
take the natural log of both sides of the equation to get:
ln(.76) = ln(e^(-.038 * t))
since ln(e^(-.038 * t) is equivalent to -.038 * t * ln(e), your equation becomes:
ln(.76) = -.038 * t * ln(e)
since ln(e) is equal to 1, your equation becomes:
ln(.76) = -.038 * t
divide both sides of this equation by -.038 to get:
ln(.76) / -.038 = t
solve for t to get t = ln(.76) / -.038 = 7.222022255
that's your answer.
it will take 7.222022255 years for the purchasing power of 1 dollar to become .76 of a dollar if the inflation rate is 3.8% per year.
you can confirm the answer is correct by replacing t in the original equation with the answer and then evaluating the equation.
you will get:
.76 = 1 * e^(-.038 * 7.222022255)
after evaluation, you will get .76 = .76.
this confirms the solution is correct.
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