A) What was the annual rate of change between 1995 and 2002?

To solve this problem, let's break down each part.

Part A: Annual Rate of Change

The value of the car depreciated from $26,000 in 1995 to $13,000 in 2002. This period spans 7 years.

The annual rate of change can be calculated using the formula for exponential decay: $V = V_0 \times (1 - r)^t$ where:

• $V$ is the final value ($13,000),
• $V_0$ is the initial value ($26,000),
• $r$ is the annual rate of depreciation, and
• $t$ is the number of years (7).

Rearrange to solve for $r$: $\frac{V}{V_0} = (1 - r)^t$ $\frac{13,000}{26,000} = (1 - r)^7$ $0.5 = (1 - r)^7$ Now, take the seventh root of both sides to isolate $1 - r$: $1 - r = (0.5)^{\frac{1}{7}}$ $r = 1 - (0.5)^{\frac{1}{7}}$ Calculating $r$ to 4 decimal places:

$r \approx 0.1041$

So, the annual rate of decrease is approximately 0.1041.

Part B: Annual Rate of Decrease in Percentage Form

To express this rate as a percentage: $r \times 100 = 0.1041 \times 100 = 10.41\%$

Thus, the annual rate of decrease is 10.41%.

Part C: Projected Value in 2006

To find the value of the car in 2006, which is 4 years after 2002 (a total of 11 years from 1995), we use the same formula: $V = V_0 \times (1 - r)^{11}$ Substitute the values: $V = 26,000 \times (1 - 0.1041)^{11}$

Calculating $V$ and rounding to the nearest $50:

$V \approx 7,500$

So, the projected value of the car in 2006 is approximately $7,500.

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Summary of Answers:

• A) The annual rate of change is 0.1041.
• B) The annual rate of decrease in percentage form is 10.41%.
• C) The value of the car in 2006 is approximately $7,500.

Would you like further details on any part?

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Here are 5 related questions to deepen understanding:

1. What formula would apply if the car's value was increasing instead?
2. How would the rate change if the depreciation occurred over 10 years instead of 7?
3. How can you confirm exponential decay versus linear depreciation?
4. How does compound interest differ from depreciation in this context?
5. What would the car’s value be after 15 years with the same rate?

Tip: Always double-check rounding requirements when presenting final answers.