If you deposit $3,590 in an account earning 5.2% interest compounded continuously, how much will the account hold in 18 months?

To calculate the amount in an account with continuously compounded interest, we use the formula:

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

Where:
- $ A $ is the amount in the account after time $ t $,
- $ P $ is the principal amount (initial deposit),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time in years,
- $ e $ is Euler's number (approximately 2.71828).

Given:
- $ P = 3,590 $,
- $ r = 5.2\% = 0.052 $,
- $ t = 18 $ months = 1.5 years.

Now let's calculate:

\[
A = 3590 \times e^{0.052 \times 1.5}
\]

I'll compute that for you.Let's manually break down the steps for the calculation:

We need to calculate:

\[
A = 3590 \times e^{0.078}
\]

First, calculate $ e^{0.078} $:
\[
e^{0.078} \approx 1.08113
\]

Now multiply by the initial deposit:
\[
A = 3590 \times 1.08113 \approx 3880.26
\]

So, the account will hold approximately **$3,880.26** after 18 months.

Would you like more details or have any other questions?

Here are 5 related questions to explore:
1. What happens if the interest rate changes to 6%?
2. How would the result differ with simple interest?
3. What if the interest is compounded annually instead of continuously?
4. How much would the account hold after 5 years?
5. What is the effect of increasing the deposit amount to $5,000?

**Tip**: Continuously compounded interest grows faster than interest compounded at regular intervals, due to the constant application of interest.

Learn how to calculate the balance of an account with an initial deposit of $3,590 and an interest rate of 5.2% compounded continuously over 18 months. This problem involves exponential functions and continuous compound interest. Suitable for Grades 10-12.

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