My investment in OHaganBooks.com stocks is losing half its value every 2 years. Find and interpret the associated decay rate. (Round your answer to one decimal place.)

To find the decay rate, we can model the value of your investment using an exponential decay function. The general form of the equation is:

\[
V(t) = V_0 \cdot e^{-kt}
\]

where:
- \( V(t) \) is the value of the investment after time \( t \),
- \( V_0 \) is the initial value of the investment,
- \( k \) is the decay rate,
- \( t \) is the time.

Since the value halves every 2 years, we know that after 2 years the value is \( \frac{V_0}{2} \). So we can set up the equation as follows:

\[
\frac{V_0}{2} = V_0 \cdot e^{-k \cdot 2}
\]

Dividing both sides by \( V_0 \) and simplifying:

\[
\frac{1}{2} = e^{-2k}
\]

Now, take the natural logarithm of both sides to solve for \( k \):

\[
\ln\left(\frac{1}{2}\right) = -2k
\]

Since \( \ln\left(\frac{1}{2}\right) = -\ln(2) \), this becomes:

\[
-\ln(2) = -2k
\]

Now solve for \( k \):

\[
k = \frac{\ln(2)}{2}
\]

Using \( \ln(2) \approx 0.6931 \):

\[
k = \frac{0.6931}{2} \approx 0.3466
\]

So the decay rate is approximately **34.7%** per year.

### Interpretation:
The decay rate of 34.7% means that the value of your investment decreases by 34.7% every year. After 2 years, your investment will be halved, consistent with the problem statement.

Do you want more details or have any questions?

Here are 5 related questions you might find useful:
1. How does the decay rate change if the value halves every 3 years instead of 2?
2. How would you compute the value of the investment after 5 years with the given decay rate?
3. Can you express the decay rate as a percentage per month instead of per year?
4. How does the exponential decay model change if the value decreases by a fixed amount every year instead of a percentage?
5. What happens to the decay rate if the investment's value reduces by 40% every 2 years instead of 50%?

**Tip:** Always use natural logarithms when dealing with exponential growth or decay problems for simplification.

Learn how to calculate the decay rate for an investment losing half its value every 2 years using the exponential decay formula. This solution explores key mathematical concepts like natural logarithms and exponential decay, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation