The formula provided for continuous compounding is:

$A = Pe^{rt}$

Where:

• $A$ is the accumulated amount,
• $P$ is the principal (initial amount invested),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years,
• $e$ is the base of the natural logarithm.

From the image:

• $P = 5500$,
• $r = 8\% = 0.08$,
• The accumulated amount $A$ is double the initial investment, so $A = 2 \times 5500 = 11000$.

We are tasked with solving for $t$, the time in years required to double the investment under continuous compounding. Let's rearrange the formula:

$A = Pe^{rt}$

Substituting the known values:

$11000 = 5500e^{0.08t}$

Dividing both sides by 5500:

$2 = e^{0.08t}$

Now, take the natural logarithm (ln) of both sides:

$\ln(2) = 0.08t$

Solve for $t$:

$t = \frac{\ln(2)}{0.08}$

Let me calculate this.The time required to double the investment under continuous compounding at an 8% annual interest rate is approximately $t \approx 8.7$ years (rounded to one decimal place).

Do you need any more details or clarifications on this?

Here are five related questions you might consider:

1. How does continuous compounding differ from annual compounding?
2. What effect would a higher interest rate have on the time required to double the investment?
3. How would the time change if the interest rate were 5% instead of 8%?
4. What is the formula for compounded interest if it's compounded annually rather than continuously?
5. How is the natural logarithm used in the context of continuous compounding?

Tip: The rule of 72 is a quick way to estimate the time required to double an investment. Divide 72 by the annual interest rate to get an approximate doubling time.