Let's solve this step-by-step:

(a) Continuous Compounding Interest Calculation

The formula for continuously compounded interest is:

$A = Pe^{rt}$

Where:

• $A$ is the future value of the investment/loan, including interest
• $P$ is the principal investment amount ($4,000 in this case)
• $r$ is the annual interest rate (29% or 0.29)
• $t$ is the time the money is invested for in years (5 years)
• $e$ is the base of the natural logarithm (approximately 2.71828)

We want to find the interest earned, which is $A - P$.

Let's calculate this value.The interest earned over 5 years with continuous compounding would be approximately $13,052.46.

(b) Finding the Equivalent Annually Compounded Interest Rate

To find the equivalent annual interest rate that corresponds to continuous compounding, we need to use the following relationship:

$(1 + r_{annual})^1 = e^{r_{continuous}}$

Where:

• $r_{continuous} = 0.29$
• $r_{annual}$ is the equivalent annually compounded rate we need to find.

Let's solve for $r_{annual}$.The equivalent annually compounded interest rate corresponding to a 29% continuously compounded rate is approximately 33.64%.

Do you have any questions or would you like further details on these calculations?

Here are 5 related questions for further understanding:

1. How does continuous compounding differ from annual compounding?
2. What would be the interest earned if the investment period were 10 years instead of 5?
3. How does changing the interest rate affect the overall interest earned with continuous compounding?
4. Can you compare continuous compounding with quarterly compounding for the same interest rate?
5. How would you calculate the future value if the interest rate were compounded monthly instead?

Tip: Continuous compounding can result in significantly more interest over time compared to other compounding methods, especially as the investment period increases.