Hannah invested $820 in an account paying an interest rate of 1.8% compounded continuously. Assuming no deposits or withdrawals are made, how much money, *to the nearest hundred dollars*, would be in the account after 16 years?

To determine how much money would be in Hannah's account after 16 years with continuous compounding, we'll use the formula:

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

where:
- $ V $ is the value of the account after $ t $ years,
- $ P $ is the initial principal ($820),
- $ r $ is the annual interest rate (1.8% or 0.018),
- $ t $ is the time in years (16 years),
- $ e $ is the base of the natural logarithm (approximately 2.71828).

Let's plug in the values:

\[
V = 820 \cdot e^{0.018 \times 16}
\]

First, calculate the exponent:

\[
0.018 \times 16 = 0.288
\]

Next, calculate $ e^{0.288} $:

\[
e^{0.288} \approx 1.334
\]

Now, multiply this by the principal:

\[
V \approx 820 \cdot 1.334 \approx 1,093.88
\]

Rounding to the nearest hundred dollars:

\[
V \approx 1,100
\]

So, after 16 years, there would be approximately **$1,100** in the account.

Would you like further details or have any questions? Here are some related questions:

1. How would the final amount change if the interest rate were 2.5% instead of 1.8%?
2. What is the difference in returns between continuous compounding and annual compounding for this investment?
3. How long would it take for the investment to double at the given interest rate?
4. What is the significance of the base $ e $ in continuous compounding?
5. How much more would Hannah have after 16 years if she invested an additional $200 at the start?

**Tip:** Always check how interest is compounded when comparing different investment options. Continuous compounding typically yields more than periodic compounding.

Explore how Hannah's $820 investment grows over 16 years with continuous compounding at an interest rate of 1.8%. Understand the formula for continuous compounding and calculate the approximate amount in Hannah's account. Discover related questions and scenarios.

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