SOLUTION: A machine depreciates in value each year by 8% of its value at the beginning of the year. After how many years will its value first be less than half its original value?

Algebra ->  Sequences-and-series -> SOLUTION: A machine depreciates in value each year by 8% of its value at the beginning of the year. After how many years will its value first be less than half its original value?      Log On


   

Question 983875: A machine depreciates in value each year by 8% of its value at the beginning of the year. After how many years will its value first be less than half its original value?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Use the formula FV = PV*(1+r)^t

where
PV = present value
FV = future value
r = rate of growth/decay. r is negative for decay
t = time in years

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FV = PV*(1+r)^t


0.5*PV = PV*(1+r)^t ... replace the future value FV with half of the present value (0.5*PV)


0.5 = (1+r)^t ... the PV terms cancel when you divide both sides by PV


0.5 = (1+(-0.08))^t ... plug in r = -0.08 to represent a decay of 8%


0.5 = (0.92)^t


(0.92)^t = 0.5


log((0.92)^t) = log(0.5) ... apply logs to both sides to help isolate t


t*log(0.92) = log(0.5)


t = log(0.5)/log(0.92)


t = 8.31295 ... use a calculator here


It takes approximately 8.31295 years for the value to be cut in half (compared to its initial value).