To solve for the time $t$ required for an investment to triple under continuous and annual compounding, let's go through the formulas required for each compounding method.

(a) Continuous Compounding

The formula for continuous compounding is: $A = Pe^{rt}$ where:

• $A$ is the amount of money accumulated after $t$ years, including interest.
• $P$ is the principal amount (initial investment).
• $r$ is the annual interest rate.
• $t$ is the time the money is invested for.

For tripling, we want $A = 3P$. Substituting this into the formula: $3P = Pe^{rt}$ Dividing both sides by $P$: $3 = e^{rt}$ Taking the natural logarithm of both sides: $\ln(3) = rt$ So: $t = \frac{\ln(3)}{r}$

Using this formula, we can calculate $t$ for each rate in the table for continuous compounding. The value of $\ln(3) \approx 1.0986$.

(b) Annual Compounding

For annual compounding, the formula is: $A = P(1 + r)^t$ Setting $A = 3P$: $3P = P(1 + r)^t$ Dividing both sides by $P$: $3 = (1 + r)^t$ Taking the natural logarithm of both sides: $\ln(3) = t \cdot \ln(1 + r)$ Solving for $t$: $t = \frac{\ln(3)}{\ln(1 + r)}$

Using this formula, we can calculate $t$ for each rate in the table for annual compounding.

I'll proceed with the calculations for each value in the table and provide the results for both continuous and annual compounding.I have displayed the table with the calculated values for both continuous and annual compounding times. Let me know if you need further explanation or have questions about these calculations.

Here are some related questions you might find helpful:

1. How does the compounding frequency affect the time required for investment growth?
2. Why is the continuous compounding time always slightly less than the annual compounding time?
3. What other methods exist for calculating interest compounding, and how do they differ?
4. What would the time be for doubling instead of tripling at the same interest rates?
5. How can we use these calculations in real-world financial planning?

Tip: Continuous compounding grows an investment faster than annual compounding due to the constant reinvestment of accrued interest, making it ideal for theoretical calculations in finance and calculus.