At the end of 1982, a company I am making up had approximately $101 million of
assets in customer accounts, and through the end of 2000 that value declined by approximately
5% per year      The CEO of the company was fired and the company was shut down when assets in customer
accounts fell below $50 million. In what month and year did that happen?
State your three decimal places answer before you interpret to month and year.


The company’s assets declined by approximately 5% per year. This represents an exponential decay problem, where the amount of assets decreases by a constant percentage over time. The general formula for exponential decay is:

\[
A(t) = A_0 \times (1 - r)^t
\]

Where:
- $A(t)$ is the amount of assets at time $t$,
- $A_0 = 101$ million is the initial amount of assets,
- $r = 0.05$ is the decay rate (5% per year),
- $t$ is the time in years.

The company is shut down when the assets fall below $50 million. So, we set up the equation:

\[
50 = 101 \times (1 - 0.05)^t
\]

We can now solve for $t$.

\[
\frac{50}{101} = (0.95)^t
\]

Taking the natural logarithm of both sides:

\[
\ln\left(\frac{50}{101}\right) = t \times \ln(0.95)
\]

Solving for $t$:

\[
t = \frac{\ln\left(\frac{50}{101}\right)}{\ln(0.95)}
\]

Let me calculate that.The calculated time $ t $ is approximately 13.707 years.

Now, since the decline started at the end of 1982, we can add this time to 1982 to determine when the assets fell below $50 million.

\[
1982 + 13.707 \approx 1995.707
\]

The decimal part, 0.707, represents a portion of the year. To find the month, we multiply this by 12:

\[
0.707 \times 12 \approx 8.484
\]

This corresponds to around the 8th month, which is August.

Thus, the assets fell below $50 million in **August 1996**.

Would you like more details, or have any questions? Here are some related questions to consider:

1. How is exponential decay applied in real-world financial situations?
2. What would happen if the decline rate was less than 5%?
3. How does compounding frequency affect exponential decay calculations?
4. How would the result change if the company started with more than $101 million in assets?
5. Can you use similar equations to predict future growth in a different scenario?

**Tip:** Exponential decay is commonly used to model depreciation, population decline, and radioactive decay.

At the end of 1982, a company I am making up had approximately $101 million of assets in customer accounts, and through the end of 2000 that value declined by approximately 5% per year. The CEO of the company was fired and the company was shut down when assets in customer accounts fell below $50 million. In what month and year did that happen? State your three decimal places answer before you interpret to month and year.

This problem explores exponential decay, where a company's assets declined by 5% per year. Using logarithmic equations, we calculate when the company's assets fell below $50 million, leading to its closure in August 1996. Suitable for students in Grades 10-12 studying logarithms and exponential decay.

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