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To solve the given problem, we use the formula for continuous compounding interest:

\[
A = P e^{rt}
\]
Where:
- $ A $ is the amount of money in the account after time $ t $.
- $ P $ is the initial principal (the initial investment).
- $ r $ is the annual interest rate (as a decimal).
- $ t $ is the time in years.
- $ e $ is the base of the natural logarithm.

### (a) Given:
- Initial investment $ P = 7100 $ dollars
- Interest rate $ r = 4.7\% = 0.047 $
- Time $ t = 13 $ years

We can calculate the amount $ A $ in the account after 13 years using the formula.

### (b) For the second part, we need to find the interest rate $ r $ for the account to reach $ A = 8100 $ dollars after 13 years, with the same initial investment of $ P = 7100 $.

I'll calculate both parts now.### Solution:
(a) After 13 years, with an interest rate of 4.7%, the account will have approximately **$13,080.14**.

(b) To reach $8,100 after 13 years, the required annual interest rate is approximately **1.01%**.

Would you like further details or clarification on any of these steps?

Here are 5 follow-up questions for further exploration:
1. How would the outcome change if the interest was compounded quarterly instead of continuously?
2. What would be the result after 20 years with the same initial investment and interest rate?
3. How much interest is earned over the 13 years with a 4.7% rate?
4. How would a higher initial investment affect the interest rate required to reach $8,100?
5. What is the formula derivation for continuously compounded interest?

**Tip:** Continuous compounding grows money faster than other compounding methods due to the infinite frequency of interest application.

Assume you invest $7100 in an account that pays interest compounded continuously. (a) If the annual interest rate were 4.7%, how much money is in the account after 13 years? (b) In order for the account to contain $8100 after 13 years, what annual interest rate would be needed?

Learn how to solve a continuous compound interest problem where $7100 is invested at an annual rate of 4.7% for 13 years. Discover how much money will accumulate and calculate the required interest rate for a target amount.

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