To solve this, we will use the formulas for continuously compounded interest and for converting continuous compounding to annually compounded interest.

Part (a): Continuous Compounding Formula

The formula for continuously compounded interest is:

$A = P e^{rt}$

Where:

• $A$ is the final amount (principal + interest).
• $P$ is the initial principal (here, $4,000).
• $r$ is the annual interest rate (29%, or 0.29).
• $t$ is the time in years (15 years).
• $e$ is the base of the natural logarithm, approximately 2.71828.

The interest earned will be:

$\text{Interest} = A - P$

Let's calculate the interest.

Part (b): Equivalent Annually Compounded Interest Rate

For part (b), we need to convert the continuously compounded rate to an equivalent annual rate. The relationship between a continuously compounded interest rate $r$ and the equivalent annual interest rate $r_a$ is given by:

$r_a = e^r - 1$

Here, $r = 0.29$. Let's calculate the equivalent annual interest rate.

I'll compute both parts now.### Part (a): The interest earned over 15 years on a deposit of $4,000 at a continuous compound rate of 29% is approximately $305,913.85.

Part (b):

The equivalent annually compounded interest rate is approximately 33.64%.

Would you like further clarification on any part of the solution?

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Here are 5 related questions to explore this topic further:

1. What would be the total amount after 20 years with the same conditions?
2. How much would a deposit of $10,000 earn over 10 years with the same rate?
3. What is the effect of time on interest if the rate stays constant?
4. How would the interest change if the rate were 25% instead of 29%?
5. How do you calculate the continuously compounded rate if only the final amount is known?

Tip: Continuous compounding grows faster than simple or annually compounded interest because it compounds every moment of the time period!