Savings   FlybynightSavings.com is offering a savings account that pays 33% compounded continuously.
(a)
How much interest would a deposit of $3,000 earn over 15 years? (Round your answer to the nearest cent.)
$ 
423675.00
 
Incorrect: Your answer is incorrect.
(b)
What is the annually compounded interest rate (in percent) that would correspond to this investment? (Round your answer to two decimal places.)
9.76
 
Incorrect: Your answer is incorrect.
 %

### Part (a) - Interest Earned on $3,000 over 15 years with Continuous Compounding

The formula for continuous compounding is given by:

\[
A = P e^{rt}
\]

Where:
- $ A $ is the final amount (principal + interest),
- $ P $ is the initial deposit (principal),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time (in years),
- $ e $ is the base of the natural logarithm (approximately 2.71828).

To find the interest earned, we first calculate the final amount and subtract the initial deposit.

**Given data### Part (a) - Interest Earned on $3,000 over 15 years with Continuous Compounding

The formula for continuous compounding is given by:

\[
A = P e^{rt}
\]

Where:
- $ A $ is the final amount (principal + interest),
- $ P $ is the initial deposit (principal),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time (in years),
- $ e $ is the base of the natural logarithm (approximately 2.71828).

To find the interest earned, we first calculate the final amount and subtract the initial deposit.

**Given **
- $ P = 3000 $ (initial deposit),
- $ r = 33\% = 0.33 $ (annual interest rate),
- $ t = 15 $ (years).

Let me calculate the total amount and the interest earned.The total amount after 15 years with continuous compounding would be approximately \$423,524.89. Therefore, the interest earned on the deposit of \$3,000 over 15 years is:

\[
\text{Interest} = A - P = 423,524.89 - 3,000 = 420,524.89
\]

So, the interest earned is **\$420,524.89**.

### Part (b) - Finding the Equivalent Annually Compounded Interest Rate

We use the relationship between the continuously compounded interest rate $ r_c $ and the annually compounded interest rate $ r_a $. The formula is:

\[
r_a = e^{r_c} - 1
\]

Where $ r_c $ is the continuously compounded rate. We can now calculate the equivalent annually compounded rate.The equivalent annually compounded interest rate corresponding to the continuous rate of 33% is approximately **39.10%**.

### Summary:
- **(a)** The interest earned on a deposit of $3,000 over 15 years with continuous compounding is **\$420,524.89**.
- **(b)** The equivalent annually compounded interest rate is **39.10%**.

Would you like further details or have any questions?

Here are some related questions to consider:
1. What happens to interest if the rate is compounded quarterly instead of continuously?
2. How would the interest change if the principal was higher?
3. What impact does a different interest rate have on continuous compounding?
4. How does compounding affect short-term versus long-term investments?
5. Can you calculate the break-even time for a different deposit amount?

**Tip:** Continuous compounding offers higher returns than traditional methods since it adds interest constantly rather than at intervals.

FlybynightSavings.com is offering a savings account that pays 33% compounded continuously. (a) How much interest would a deposit of $3,000 earn over 15 years? (b) What is the annually compounded interest rate that would correspond to this investment?

Calculate the interest earned on a $3,000 deposit with 33% continuous compounding over 15 years. Learn how to use the continuous compounding formula and convert it to an annual compounded interest rate. Ideal for advanced high school and college students.

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