Question 1145684
the purchase price is 32,145.
it depreciates at the rate of 7.5% per year.
the formula to use is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is then number of time periods.


the time periods are years.


in your problem:
f = what you wand to find
p = 32,145
r = -7.5% = -.075
n = 5.5


formula becomes f = 32,145 * (1 - .075) ^ 5.5
solve for f to get f = 20,935.96387
that's the value after 5.5 years.


to determine how long until the value drops to 10,000, the same formula is used.
f = 10,000
p = 32,145
r = -7.5% per year = -.075
n = what you want to find.


the formula becomes 10,000 = 32,145 * (1 - .075) ^ n
divide both sides of this equation by 32,145 to get:
10,000 / 32,145 = (1 - .075) ^ n
take the log of both sides of this equation to get:
log(10,000 / 32,145) = log((1 - .075) ^ n)
since log((1 - .075) ^ n) is equal to n * log(1 - .075), you get:
log(10,000 / 32,145) = n * log(1 - .075)
divide both sides of this equation by log(1 - .075) to get:
log(10,000 / 32,145) / log(1 - .075) = n
solve for n to get n = 14.97753639.
confirm by replacing n in the original equation with that to get:
10,000 = 32,145 * (1 - .075) ^ 14.97753639.
evaluate that equation to get:
10,000 = 10,000
this confirms the value of n is correct.


your solution is that it will take 14.97753639 years for the original value to become 10,000 dollars.