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This problem involves continuous compounding, which is modeled by the formula:

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

Where:
- $ A $ is the amount of money in the account after time $ t $,
- $ P $ is the initial deposit (what we need to find),
- $ r $ is the interest rate (in decimal form),
- $ t $ is the time in years,
- $ e $ is Euler's number (approximately 2.71828).

We are given:
- $ A = 7789 $,
- $ r = 5.33\% = 0.0533 $,
- $ t = 16 $ years.

We need to find $ P $, so we rearrange the formula:

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

Substituting the values into the formula:

\[
P = \frac{7789}{e^{0.0533 \times 16}}
\]

I'll calculate this now.The initial deposit made into the account was \$3,319.83.

Would you like further details or have any questions about the solution?

Here are five related questions to deepen your understanding:

1. How would the initial deposit change if the interest rate was 6% instead of 5.33%?
2. What is the effect of continuous compounding compared to annual compounding for the same interest rate?
3. How much would the account hold after 20 years instead of 16?
4. What would happen to the final amount if the initial deposit remained the same but the interest was compounded quarterly instead of continuously?
5. How is the continuous compounding formula derived from the standard compound interest formula?

**Tip:** When working with continuous compounding, always ensure you convert the interest rate into decimal form and multiply it by time before using the exponential function.

Suppose an account containing $7,789 was invested at 5.33% compounded continuously for 16 years. What was the initial deposit made into the account?

Calculate the initial deposit required for an account that grows to $7,789 with a 5.33% interest rate compounded continuously over 16 years. Learn how to apply the continuous compounding formula.

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