We are given a depreciation rate of 17% annually, until the value reaches $500.

(a) The car is worth 25,000 originally. The formula for compound depreciation is A=P(1-r)^t where A is the amount the car is worth at time t, P is the original price, r is the annual depreciation...

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We are given a depreciation rate of 17% annually, until the value reaches $500.

(a) The car is worth 25,000 originally. The formula for compound depreciation is A=P(1-r)^t where A is the amount the car is worth at time t, P is the original price, r is the annual depreciation rate, and t is the time in years.

Here A=500,P=25000,r=.17 and we are solving for t:

500=25000(1-.17)^t

.02=.83^t Take a logarithm (I will use the natural logarithm but it does not matter) of each side.

ln(.02)=ln(.83^t) Use a property of logs:

ln(.02)=tln(.83) So

t=[ln(.02)]/[ln(.83)] or t is approximately 20.995 or 21 years.

The car will be worth $500 in 2034.

(b)Find the depreciation rate so that a car loses 2/3 of its value in 7 years.

Use the same formula with A=2/3, P=1, r unknown, and t=7:

2/3=(1-r)^7 Take the 7th root of both sides (or raise both sides to the 1/7 power) to get:

(2/3)^(1/7)=1-r so r=1-(2/3)^(1/7). Thus r is approximately .0563.

r is about 5.63%

(c) P=13,995.36, A=4477.31, r is unknown and t=6 (2015-2009)

4477.31=13995.36(1-r)^6

.3199=(1-r)^6 Take the 6th root of both sides:

.827=1-r ==> r is about .173 so r is 17.3%.

Now we have A=13995.36, P is unknown, the depreciation rate r is .173 and t is 5 (2009-2004)

13995.36=P(1-.173)^5

13995.36=P(.38684)

P is about 36179 .

The original owner paid $36,179

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