Question

A car was valued at $35,000 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?
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 2009 ?
value = $ Round to the nearest 50 dollars.

Answer 1

To solve these questions, we'll start by determining the annual rate of change between 1995 and 2004 for the value of the car.

### A) Annual Rate of Change

1. Let \( V_0 \) be the initial value in 1995, which is $35,000.
2. Let \( V_t \) be the value in 2004, which is $13,000.
3. The time period from 1995 to 2004 is 9 years.

The formula for exponential decay is:
\[
V_t = V_0 \times (1 + r)^t
\]

Where:
- \( V_t \) is the value at time \( t \)
- \( V_0 \) is the initial value
- \( r \) is the annual rate of change (as a decimal)
- \( t \) is the number of years

We can rearrange the formula to solve for \( r \):
\[
r = \left(\frac{V_t}{V_0}\right)^{\frac{1}{t}} - 1
\]

Plugging in the values:
\[
r = \left(\frac{13000}{35000}\right)^{\frac{1}{9}} - 1
\]

Calculating \( \frac{13000}{35000} \):
\[
\frac{13000}{35000} \approx 0.3714285714
\]

Now calculate \( 0.3714285714^{\frac{1}{9}} \):
\[
0.3714285714^{\frac{1}{9}} \approx 0.7360040824
\]

Subtracting 1 gives us:
\[
r \approx 0.7360040824 - 1 \approx -0.2639959176
\]

Rounding to 4 decimal places:
\[
r \approx -0.2640
\]

### B) Rate in Percentage Form

To convert the decimal to percentage, we multiply by 100:
\[
r \approx -0.2640 \times 100 \approx -26.40\%
\]

### C) Value in 2009

Now we need to calculate the value in 2009, which is 5 years after 2004. We'll use the same decay formula:

\[
V_{2009} = V_{2004} \times (1 + r)^t
\]
Where:
- \( V_{2004} = 13000 \)
- \( r \approx -0.2640 \)
- \( t = 5 \)

Calculating:
\[
V_{2009} = 13000 \times (1 - 0.2640)^5
\]

Calculating \( (0.7360)^5 \):
\[
(0.7360)^5 \approx 0.2067986836
\]

Now, multiplying:
\[
V_{2009} = 13000 \times 0.2067986836 \approx 2688.373886
\]

Rounding to the nearest $50:
\[
V_{2009} \approx 2700
\]

### Summary of Answers:
- A) \( r = -0.2640 \)
- B) \( r = -26.40\% \)
- C) Value in 2009: $2700