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.