Find a formula for the exponential function V=h(t)
 that gives the value of an item initially worth $5000 that loses half its value every 5 years.

h(t)=
 
 dollars

The value of an item that loses half its value every 5 years can be modeled using an exponential decay function. 

The general form of an exponential decay function is:

\[
V(t) = V_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T}}
\]

Where:
- $ V(t) $ is the value of the item at time $ t $,
- $ V_0 $ is the initial value of the item,
- $ T $ is the time it takes for the value to halve (in this case, 5 years),
- $ t $ is the time in years.

For this problem:
- $ V_0 = 5000 $ (initial value of the item),
- $ T = 5 $ (since it loses half its value every 5 years).

Thus, the formula for the value of the item after $ t $ years is:

\[
h(t) = 5000 \cdot \left(\frac{1}{2}\right)^{\frac{t}{5}}
\]

This gives the value of the item $ h(t) $ in dollars after $ t $ years.

Would you like further explanation or details?

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Here are 5 related questions you might consider:
1. How would the formula change if the item loses a different fraction of its value every 5 years?
2. What is the value of the item after 10 years?
3. How does the graph of this function look over time?
4. Can this model apply to other real-life depreciation scenarios, like car values?
5. How would you calculate the time it takes for the value to drop to a specific amount?

**Tip:** Exponential decay models are useful for depreciation, population decay, and radioactive decay. Understanding how to adjust parameters helps you apply it in various fields.

Find a formula for the exponential function V=h(t) that gives the value of an item initially worth $5000 that loses half its value every 5 years.

Learn how to calculate the value of an item that loses half its value every 5 years using an exponential decay function. Starting with an initial value of $5000, this detailed explanation covers key mathematical concepts like exponential functions and decay. Suitable for high school students in Grades 9-12.

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