A car was valued at $32,000 in the year 1995. The value depreciated to $13,000 by the year 2000.

A) What was the annual rate of change between 1995 and 2000?
 = 
 Round the rate of decrease to 4 decimal 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 2004 ?
value = $ 
 Round to the nearest 50 dollars.

### Part A: Annual Rate of Change

To find the annual rate of change, we assume exponential decay of the form:

\[
V(t) = V_0 \cdot (1 - r)^t
\]

Where:
- $V(t)$ is the value of the car after $t$ years,
- $V_0$ is the initial value of the car,
- $r$ is the annual rate of decrease (which we need to find),
- $t$ is the time in years.

In 1995, $V_0 = 32,000$, and in 2000 ($t = 5$ years), the value is $13,000$.

The equation becomes:

\[
13,000 = 32,000 \cdot (1 - r)^5
\]

To solve for $r$, we first divide both sides by 32,000:

\[
\frac{13,000}{32,000} = (1 - r)^5
\]

\[
0.40625 = (1 - r)^5
\]

Now, take the 5th root of both sides to get rid of the exponent:

\[
1 - r = \sqrt[5]{0.40625}
\]

Let me compute the 5th root and solve for $r$.The annual rate of decrease, rounded to 4 decimal places, is **0.1649**.

### Part B: Percentage Form

To express the rate in percentage form, multiply by 100:

\[
0.1649 \times 100 = 16.49\%
\]

Thus, the annual rate of decrease is **16.49%**.

### Part C: Value of the Car in 2004

To find the value of the car in 2004, we use the same exponential decay model. The number of years between 1995 and 2004 is $t = 9$. The formula is:

\[
V(9) = 32,000 \cdot (1 - 0.1649)^9
\]

Let me compute the value of the car in 2004.The value of the car in the year 2004, rounded to the nearest 50 dollars, is **$6,300**.

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Would you like any further details or clarifications?

Here are 5 related questions you may want to explore:
1. How would the value of the car change if the rate of depreciation were 20% annually?
2. How long would it take for the car's value to drop below $5,000 with the same depreciation rate?
3. What is the half-life of the car's value given the same depreciation rate?
4. How does the exponential depreciation model compare to linear depreciation?
5. How would a sudden increase in the depreciation rate impact the car's value over the next 5 years?

**Tip:** Exponential decay models are commonly used for depreciation because they better capture the faster value drops earlier in an asset's lifetime.

A car was valued at $32,000 in the year 1995. The value depreciated to $13,000 by the year 2000.

A) What was the annual rate of change between 1995 and 2000?
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 2004?

Learn how to calculate the annual depreciation rate of a car valued at $32,000 in 1995 and $13,000 in 2000. Find the percentage rate and predict the car's value in 2004, using exponential decay formulas. Suitable for Grades 9-12.

Math Problem Statement

A) What was the annual rate of change between 1995 and 2000?

B) What is the correct answer to part A written in percentage form?

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