How many years will it take $ 6 comma 000$6,000 to grow to $ 11 comma 600$11,600 if it is invested at 3.75 %3.75% compounded​ continuously?
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Part 1
enter your response here years ​(Round to two decimal​ places.)

To solve this, we will use the formula for continuous compounding, which is given by:

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

Where:
- $ A $ is the final amount (11,600),
- $ P $ is the initial principal (6,000),
- $ r $ is the annual interest rate (3.75% or 0.0375),
- $ t $ is the time in years,
- $ e $ is the base of the natural logarithm.

We need to solve for $ t $. First, rearrange the formula to isolate $ t $:

\[
t = \frac{\ln(A/P)}{r}
\]

Now, substitute the given values into the equation:

\[
t = \frac{\ln(11,600 / 6,000)}{0.0375}
\]

Let's calculate that.

It will take approximately 17.58 years for $6,000 to grow to $11,600 if it is invested at an interest rate of 3.75% compounded continuously.

Would you like more details or have any questions?

Here are 5 related questions:
1. What if the interest rate was 5%? How long would it take?
2. How much would $6,000 grow to in 10 years at 3.75% continuous compounding?
3. What is the difference between continuous and annual compounding in terms of growth?
4. Can you derive the continuous compounding formula?
5. How does changing the principal amount affect the growth period?

**Tip:** Continuous compounding allows money to grow at the fastest rate possible for a given interest rate.

How many years will it take $6,000 to grow to $11,600 if it is invested at 3.75% compounded continuously?

Learn how to calculate the time required for $6,000 to grow to $11,600 with continuous compounding at an interest rate of 3.75%. This detailed guide covers key exponential growth concepts, the continuous compounding formula, and how to use natural logarithms. Suitable for students in Grades 10-12.

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