A quantity with an initial value of 8100 decays continuously at a rate of 0.95% per year. What is the value of the quantity after 96 months, to the nearest hundredth?

The formula to model continuous decay is given by:

\[
A = A_0 e^{-rt}
\]

Where:
- \(A\) is the final amount.
- \(A_0\) is the initial amount.
- \(r\) is the decay rate (expressed as a decimal).
- \(t\) is the time period in years.
- \(e\) is the base of the natural logarithm, approximately 2.71828.

### Given:
- \(A_0 = 8100\)
- \(r = 0.0095\) (since 0.95% decay rate is 0.0095 as a decimal)
- \(t = 96\) months \(= \frac{96}{12} = 8\) years.

Now, we can plug these values into the formula:

\[
A = 8100 e^{-0.0095 \times 8}
\]

Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate it manually as follows:

1. Multiply the decay rate \( r \) by the time in years \( t \) (which is 8).
2. Take the negative of that product and use it as the exponent of \( e \), the base of the natural logarithm.
3. Multiply the result by the initial value of 8100 to get the final quantity.

Would you like more clarification or any other details on how this works?

Here are some related questions to think about:
1. What is the formula for exponential growth?
2. How does the decay constant affect the final value?
3. What happens if the decay rate is increased to 1.5%?
4. How would this decay formula change if the decay was not continuous but discrete?
5. How long will it take for the quantity to halve at this decay rate?

**Tip:** Continuous decay or growth uses the base \( e \), which is essential for natural processes like radioactive decay and population growth.

Calculate the exponential decay of a quantity starting at 8100 with a continuous decay rate of 0.95% per year over a period of 96 months. This step-by-step guide covers the exponential decay formula and the role of the natural logarithm. Perfect for students in Grades 9-12.

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