A car was valued at $29,000 in the year 1993. The value depreciated to $10,000 by the year 2004.

A) What was the annual rate of change between 1993 and 2004?
 = 
 Round the rate of decrease to 4 decimal places.

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

To solve this problem, we will use the formula for exponential decay, since the car's value is decreasing over time. 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 rate of decay per year,
- $ t $ is the number of years.

### Given:
- Initial value $ V_0 = 29,000 $,
- Final value $ V(t) = 10,000 $,
- Time span $ t = 2004 - 1993 = 11 $ years.

We need to find the annual rate of depreciation $ r $.

### Step 1: Set up the equation

\[
10,000 = 29,000 \times (1 - r)^{11}
\]

### Step 2: Solve for $ 1 - r $

First, divide both sides of the equation by 29,000:

\[
\frac{10,000}{29,000} = (1 - r)^{11}
\]

\[
\frac{10}{29} \approx (1 - r)^{11}
\]

Now, take the 11th root of both sides:

\[
(1 - r) = \left( \frac{10}{29} \right)^{\frac{1}{11}}
\]

Using a calculator:

\[
(1 - r) \approx 0.8017
\]

### Step 3: Solve for $ r $

\[
r = 1 - 0.8017
\]

\[
r \approx 0.1983
\]

So, the annual rate of depreciation is approximately **0.1983** (to four decimal places).

### Part B: Convert the rate into percentage form

To express this rate as a percentage, multiply by 100:

\[
r \times 100 \approx 19.83\%
\]

### Final Answers:
- A) The annual rate of change is **0.1983**.
- B) The rate of decrease in percentage form is **19.83%**.

Would you like more details on this, or do you have any questions?

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Here are 5 related questions:
1. What happens to the value of the car if the rate of depreciation decreases?
2. How would the formula change if the car's value increases over time?
3. Can the rate of depreciation be constant for all assets over time?
4. What is the value of the car after 15 years at the same depreciation rate?
5. How would the problem change if the car's value was given after 20 years instead of 11?

**Tip**: When dealing with percentage changes, always remember to multiply the decimal rate by 100 to express it as a percentage.

A car was valued at $29,000 in the year 1993. The value depreciated to $10,000 by the year 2004. A) What was the annual rate of change between 1993 and 2004? B) What is the correct answer to part A written in percentage form?

Discover how to calculate the annual depreciation rate of a car that decreased in value from $29,000 in 1993 to $10,000 in 2004 using the exponential decay formula. This example explains step-by-step how to find the rate of depreciation and express it in percentage form. Suitable for students in Grades 10-12.

Math Problem Statement

A) What was the annual rate of change between 1993 and 2004?

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

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