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The problem states that your investment in OHaganBooks.com stocks loses half of its value every 3 years. This describes an exponential decay scenario, where the value of the stock decreases over time by a fixed proportion.

### Step 1: General Exponential Decay Formula
The general formula for exponential decay is:
\[
A(t) = A_0 \cdot e^{-kt}
\]
Where:
- \( A(t) \) is the value of the investment after time \( t \),
- \( A_0 \) is the initial value of the investment,
- \( k \) is the decay constant (which we need to find),
- \( t \) is the time in years.

### Step 2: Use the Given Information
We know that the investment loses half of its value every 3 years. So when \( t = 3 \), the value becomes half of the original, i.e., \( A(3) = \frac{1}{2} A_0 \).

Plugging this into the exponential decay formula:
\[
\frac{1}{2} A_0 = A_0 \cdot e^{-3k}
\]

### Step 3: Solve for \( k \)
Dividing both sides by \( A_0 \) (assuming \( A_0 \neq 0 \)) gives:
\[
\frac{1}{2} = e^{-3k}
\]

Take the natural logarithm (ln) of both sides:
\[
\ln \left( \frac{1}{2} \right) = -3k
\]

Since \( \ln \left( \frac{1}{2} \right) = -\ln(2) \), this simplifies to:
\[
-\ln(2) = -3k
\]

Now solve for \( k \):
\[
k = \frac{\ln(2)}{3}
\]

Using \( \ln(2) \approx 0.693 \):
\[
k \approx \frac{0.693}{3} \approx 0.231
\]

### Step 4: Interpretation of the Decay Rate
The decay constant \( k \approx 0.231 \) means that the investment is decaying at a rate of about 23.1% per year. This reflects the continuous rate at which the stock's value is reducing.

Would you like more details on how to use this decay rate in other financial contexts?

### Relative Questions:
1. How would the decay constant change if the investment lost half its value every 5 years instead?
2. How can we use this decay rate to predict the value of the stock after 10 years?
3. What is the half-life formula and how is it derived from exponential decay?
4. How does continuous decay differ from discrete forms of depreciation in investments?
5. What are real-world examples where such exponential decay is commonly observed?

**Tip:** To quickly estimate decay over time, remember the rule of 72, which approximates the time it takes for an investment to halve or double.

My investment in OHaganBooks.com stocks is losing half its value every 3 years. Find and interpret the associated decay rate.

Learn how to calculate the exponential decay rate for an investment that loses half its value every 3 years. This problem covers key concepts like continuous decay, logarithms, and the exponential decay formula. Suitable for Grades 10-12.

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