Question 1186430: A car was valued at $27,000 in the year 1994. The value depreciated to $12,000 by the year 2001.
A) What was the annual rate of change between 1994 and 2001?
r
=
Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r
=
%.
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2006 ?
value = $
Round to the nearest 50 dollars.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! A) To find the annual rate of change, we can use the formula for exponential decay:
```
Final Value = Initial Value * (1 - r)^t
```
Where:
* Final Value = $12,000
* Initial Value = $27,000
* r = annual rate of change (what we want to find)
* t = number of years = 2001 - 1994 = 7
Let's plug in the values and solve for r:
```
12000 = 27000 * (1 - r)^7
(12000/27000) = (1 - r)^7
(4/9) = (1 - r)^7
(4/9)^(1/7) = 1 - r
r = 1 - (4/9)^(1/7)
r ≈ 0.1094
```
Therefore, the annual rate of change is approximately **0.1094**.
B) To express the rate in percentage form, we simply multiply by 100:
```
r ≈ 0.1094 * 100 = 10.94%
```
Therefore, the annual rate of change is approximately **10.94%**.
C) To find the value in 2006, we can use the same formula, but with t = 2006 - 1994 = 12:
```
Value in 2006 = 27000 * (1 - 0.1094)^12
Value in 2006 ≈ 6715.64
```
Rounding to the nearest 50 dollars, the value of the car in 2006 will be approximately **$6700**.
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