A car was valued at $26,000 in the year 1995. The value depreciated to $13,000 by the year 2004.

A) What was the annual rate of change between 1995 and 2004? Round the rate of decrease to 4 decimals places
B) What is the correct answer to part A written in percentage form?
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2009 ? Round to the nearest 50 dollars.

A) 13000 = 26000 (1 - r)^9

B) r * 100%

C) v = 26000 (1 - r)^14

say value at year x+1 = d * value at year x

then value at year x+ n = d^n * value at year x
or
value at year 2004 = d^(2004 - 1995) * value at year 1995
so
d^9 = 13000/26000 = 0.5
9 log d = log 0.5
log d = log 0.5 / 9 = -0.301 / 9 = -0.03345
d = 10^-0.03345
d = 0.9259 or decreases by .07412 of its value per year
decrease by 100 - 92.59 = 7.412 % per year

value at year 2009 = d^(2009 - 1995) * value at year 1995
= 0.9259^14 * 26000 = 0.3403 * 26,000 = 8,848

A. V = Vo(1-r)^9 = 13,000

26,000(1-r)^9 = 13,000
(1-r)^9 = 0.5
1-r = 0.9259
r = 0.0741.
00
B. .r = 7.41 %.

C. V = 26,000(1-0.0741)^14 =

4

5

To find the annual rate of change, or the rate of depreciation between 1995 and 2004, we need to use the formula:

Rate = (Final Value - Initial Value) / (Number of Years)

A) Applying the formula:
Initial Value = $26,000
Final Value = $13,000
Number of Years = 2004 - 1995 = 9 years

Rate = ($13,000 - $26,000) / 9
Rate = -$13,000 / 9
Rate = -$1444.44

Rounded to 4 decimal places, the annual rate of change (or rate of depreciation) is approximately -$1444.4400.

B) To express this rate in percentage form, we divide the annual rate of change by the initial value and multiply by 100:

Percentage Rate = (Rate / Initial Value) * 100
Percentage Rate = (-$1444.44 / $26,000) * 100
Percentage Rate ≈ -5.56% (rounded to 2 decimal places)

So, the correct answer to part A written in percentage form is approximately -5.56%.

C) Assuming the car value continues to drop by the same percentage, we can calculate the value in the year 2009 by multiplying the original value ($26,000) by the percentage rate (-5.56%) for each year from 2004 to 2009.

Value in 2004 = $13,000

Value in 2005 = Value in 2004 * (1 + Percentage Rate)
Value in 2005 = $13,000 * (1 - 0.0556) ≈ $12,430

Value in 2006 = Value in 2005 * (1 - 0.0556)
Value in 2006 ≈ $12,430 * (1 - 0.0556) ≈ $11,734

Continuing this process for each subsequent year:

Value in 2007 ≈ $11,734 * (1 - 0.0556) ≈ $11,053
Value in 2008 ≈ $11,053 * (1 - 0.0556) ≈ $10,390
Value in 2009 ≈ $10,390 * (1 - 0.0556) ≈ $9,741

Rounded to the nearest $50, the value in the year 2009 will be approximately $9,750.