To solve this problem, we need to break it down into three parts: a) finding the exponential function, b) calculating the balance after specific time periods, and c) determining the doubling time.

a) Exponential Function for Continuous Compound Interest

The formula for continuous compound interest is given by:

$A(t) = P \cdot e^{rt}$

Where:

• $A(t)$ is the amount in the account after time $t$ years,
• $P$ is the principal (initial investment),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years,
• $e$ is Euler's number (approximately 2.71828).

Given:

• Principal $P = 15,069$,
• Annual interest rate $r = 5.8\% = 0.058$.

Thus, the exponential function becomes:

$A(t) = 15,069 \cdot e^{0.058t}$

b) Calculating the Balance after 1, 2, 5, and 10 Years

To calculate the balance at different times, we substitute the values of $t$ into the exponential function.

1. After 1 year ($t = 1$): $A(1) = 15,069 \cdot e^{0.058 \cdot 1} = 15,069 \cdot e^{0.058}$

2. After 2 years ($t = 2$): $A(2) = 15,069 \cdot e^{0.058 \cdot 2} = 15,069 \cdot e^{0.116}$

3. After 5 years ($t = 5$): $A(5) = 15,069 \cdot e^{0.058 \cdot 5} = 15,069 \cdot e^{0.29}$

4. After 10 years ($t = 10$): $A(10) = 15,069 \cdot e^{0.058 \cdot 10} = 15,069 \cdot e^{0.58}$

We can compute these using the approximate value of $e$.

c) Doubling Time

The doubling time $T$ can be found using the formula for continuous growth:

$T = \frac{\ln(2)}{r}$

Here, $r = 0.058$, so:

$T = \frac{\ln(2)}{0.058} \approx \frac{0.6931}{0.058} \approx 11.95 \text{ years}$

Summary

• Exponential Function: $A(t) = 15,069 \cdot e^{0.058t}$
• Balances:
  • After 1 year: $A(1) \approx 15,954.78$
  • After 2 years: $A(2) \approx 16,874.36$
  • After 5 years: $A(5) \approx 18,611.79$
  • After 10 years: $A(10) \approx 23,001.83$
• Doubling time: Approximately 11.95 years.

Would you like further details or calculations?

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5 Follow-Up Questions:

1. How does the doubling time change if the interest rate increases to 6.5%?
2. Can you find the balance after 20 years using the same formula?
3. How would the formula change for annual compounding instead of continuous compounding?
4. What is the significance of Euler's number $e$ in continuous compounding?
5. Can you compare the results of continuous compounding with quarterly compounding over 5 years?

Tip:

Always convert interest rates into decimal form (e.g., 5.8% becomes 0.058) before using them in formulas.