Let's analyze and solve each part of the problem.

Part (a): Calculating the Interest for Continuous Compounding

The formula for continuously compounded interest is:

$A = Pe^{rt}$

where:

• $A$ is the future amount,
• $P$ is the initial principal (the amount of money deposited),
• $e$ is Euler's number (approximately 2.71828),
• $r$ is the interest rate (as a decimal),
• $t$ is the time in years.

For this problem:

• $P = 2000$,
• $r = 0.28$ (28% as a decimal),
• $t = 15$.

Let me calculate $A$ first and then find the interest, which is $A - P$.

Part (b): Finding the Equivalent Annual Compounded Interest Rate

To find the equivalent annual rate $r_{annual}$, we need to set the continuously compounded formula equal to the formula for annual compounding:

$Pe^{rt} = P(1 + r_{annual})^t$

Simplifying:

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

Solving for $r_{annual}$:

$1 + r_{annual} = (e^{rt})^{1/t}$

$r_{annual} = (e^{rt})^{1/t} - 1$

I will compute these values now.### Results

• Part (a): The interest earned on a deposit of $2,000 over 15 years with a 28% continuously compounded interest rate is $131,372.66.

• Part (b): The equivalent annually compounded interest rate is 32.31%.

Would you like further details or have any questions?

Related Questions

1. How does the calculation change if the compounding is done monthly instead of continuously?
2. What would be the interest earned if the deposit was $5,000 instead?
3. How would the final amount change if the investment period is shortened to 10 years?
4. What happens to the equivalent annual interest rate if the continuous rate is 35% instead of 28%?
5. How can the formula be used to determine the required initial deposit for a target future amount?

Tip

For long-term investments, continuously compounded interest grows faster than other compounding methods due to the continuous accumulation of interest.