 new car is purchased for dollar sign, 18, comma, 000$18,000 and over time its value depreciates by one half every 4 years. What is the value of the car 17 years after it was purchased, to the nearest hundred dollars?

To find the value of the car 17 years after it was purchased, we can use the formula for exponential decay, which is given by:

\[
V(t) = P \times \left(\frac{1}{2}\right)^{\frac{t}{T}}
\]

where:
- $ V(t) $ is the value of the car at time $ t $ years,
- $ P $ is the initial value of the car (which is $18,000),
- $ T $ is the time it takes for the car to depreciate by half (which is 4 years in this case),
- $ t $ is the number of years after the purchase.

Given:
- $ P = 18,000 $ dollars,
- $ T = 4 $ years,
- $ t = 17 $ years.

We can substitute these values into the formula:

\[
V(17) = 18,000 \times \left(\frac{1}{2}\right)^{\frac{17}{4}}
\]

First, calculate the exponent:

\[
\frac{17}{4} = 4.25
\]

Then calculate the decay factor:

\[
\left(\frac{1}{2}\right)^{4.25}
\]

Finally, multiply by the initial value:

\[
V(17) = 18,000 \times \left(\frac{1}{2}\right)^{4.25}
\]

Let's compute this.The value of the car 17 years after it was purchased is **$900** when rounded to the nearest hundred dollars.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the value change if the depreciation rate was different?
2. What would be the value after 20 years instead of 17?
3. How does the depreciation formula change if the value decreases by a fixed percentage each year instead of halving every 4 years?
4. What is the half-life of an investment that depreciates by 25% every 5 years?
5. How would the problem change if the car's value depreciated continuously instead of every 4 years?

**Tip:** When dealing with exponential decay, understanding the half-life is crucial for quickly estimating the value after a given period.

New car is purchased for $18,000 and over time its value depreciates by one half every 4 years. What is the value of the car 17 years after it was purchased, to the nearest hundred dollars?

Calculate the value of a car purchased for $18,000 after 17 years, considering its value depreciates by half every 4 years. Learn how to use the exponential decay formula to determine the car's value, rounded to the nearest hundred dollars.

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