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Let's break down the problem and solve it step-by-step.

### Given:
- Initial value of the car in 1993 = $26,000
- Value of the car in 2006 = $14,000
- Time period = 2006 - 1993 = 13 years

We assume the depreciation follows an exponential decay model:
\[
V(t) = V_0 \times (1 - r)^t
\]
Where:
- $V(t)$ is the value of the car after time $t$,
- $V_0$ is the initial value,
- $r$ is the annual rate of depreciation,
- $t$ is the time in years.

For part (A), we need to find $r$ (the rate of depreciation).

#### Step 1: Using the exponential decay model
We know:
\[
14,000 = 26,000 \times (1 - r)^{13}
\]

Divide both sides by 26,000:
\[
\frac{14,000}{26,000} = (1 - r)^{13}
\]
\[
0.5385 = (1 - r)^{13}
\]

#### Step 2: Solve for $r$
To solve for $r$, take the 13th root of both sides:
\[
(1 - r) = (0.5385)^{\frac{1}{13}}
\]

Now calculate the right-hand side.

The annual rate of decrease $r$ is approximately 0.0465 (rounded to four decimal places).

### Part A: 
\[
r = 0.0465
\]

### Part B: Convert $r$ to percentage form:
To express $r$ as a percentage, multiply by 100:
\[
r = 0.0465 \times 100 = 4.6502\%
\]

### Part B:
\[
r = 4.6502\%
\]

### Part C: Predicting the value in 2009

Using the same formula, we can predict the value of the car in 2009 (16 years after 1993). The formula is:
\[
V(16) = 26,000 \times (1 - 0.0465)^{16}
\]

Let's calculate the value in 2009.

### Part C:
The estimated value of the car in 2009 is approximately **$12,150**, rounded to the nearest 50 dollars.

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Let me know if you would like further details on any part of this solution. 

Here are 5 related questions to explore:
1. How would the calculation change if the car's value dropped at a different rate every year?
2. What is the total depreciation amount from 1993 to 2006?
3. Can you derive the value of the car in a different year, say 2010, using the same formula?
4. What is the compound annual growth rate (CAGR) formula and how is it related to the formula used here?
5. How would inflation affect the real value of the car over this period?

**Tip:** Always ensure to apply the correct time period when using exponential growth or decay models for financial calculations.

A car was valued at $26,000 in the year 1993. The value depreciated to $14,000 by the year 2006. A) What was the annual rate of change between 1993 and 2006? 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?

Learn how to calculate the annual depreciation rate of a car valued at $26,000 in 1993, which dropped to $14,000 by 2006. This problem involves exponential decay and percentage calculations, predicting the value in 2009 as well. Suitable for high school students (Grades 9-12).

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