FlybynightSavings.com is offering a savings account that pays 32% compounded continuously.
(a)
How much interest would a deposit of $2,000 earn over 10 years? (Round your answer to the nearest cent.)
$
(b)
What is the annually compounded interest rate (in percent) that would correspond to this investment? (Round your answer to two decimal places.)
%

Let's solve both parts of the problem using the given continuous compounding interest formula:

The formula for continuously compounded interest is:

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

Where:
- $ A $ is the final amount,
- $ P $ is the principal (initial deposit),
- $ 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).

### Part (a): 
We are asked to find how much interest a deposit of $2,000 will earn in 10 years at a rate of 32% compounded continuously.

Here:
- $ P = 2000 $,
- $ r = 0.32 $ (32% as a decimal),
- $ t = 10 $.

The interest earned is the difference between the final amount $ A $ and the initial deposit $ P $. So we will calculate:

\[
A = 2000 \cdot e^{(0.32 \cdot 10)}
\]

and the interest earned will be:

\[
\text{Interest} = A - 2000
\]

I'll calculate that now.The interest earned on a deposit of $2,000 over 10 years at a continuously compounded rate of 32% is **$47,065.06**.

### Part (b):
Now, let's find the equivalent annual interest rate that would correspond to this investment if compounded annually. 

For this, we'll use the formula for annual compounding:

\[
A = P \cdot \left(1 + r_{annual}\right)^t
\]

We already know the final amount $ A $ from part (a), so we set it equal to the result from continuous compounding and solve for $ r_{annual} $:

\[
A = 2000 \cdot \left(1 + r_{annual}\right)^{10}
\]

This simplifies to:

\[
r_{annual} = \left(\frac{A}{2000}\right)^{1/10} - 1
\]

I'll compute this now.The annually compounded interest rate equivalent to the continuously compounded rate of 32% is **37.71%**.

### Summary:
- (a) The interest earned on a deposit of $2,000 over 10 years at 32% compounded continuously is **$47,065.06**.
- (b) The equivalent annually compounded interest rate is **37.71%**.

Do you want more details or have any questions about these calculations?

Here are 5 related questions to expand on:
1. How would the result change if the time period was reduced to 5 years?
2. What would be the interest earned if the rate was 25% compounded continuously?
3. How does the concept of continuous compounding differ from simple compounding?
4. Can you derive the equivalent monthly compounded interest rate?
5. What is the effect of compounding frequency on interest earned?

**Tip:** Continuous compounding results in slightly higher returns compared to other compounding methods because it assumes the principal is growing constantly.

FlybynightSavings.com is offering a savings account that pays 32% compounded continuously.
(a) How much interest would a deposit of $2,000 earn over 10 years? (Round your answer to the nearest cent.)
(b) What is the annually compounded interest rate (in percent) that would correspond to this investment? (Round your answer to two decimal places.)

Solve how much interest a $2,000 deposit earns over 10 years at 32% compounded continuously. Learn the continuous compounding formula and find the equivalent annual interest rate. Suitable for students in Grades 10-12 studying exponential growth and compound interest.

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