The problem asks us to find the number of years it takes for an initial investment of $500 to grow to $995 at an interest rate of 8% compounded continuously. We need to round the answer to the nearest whole number.

1. Problem Description

The problem asks us to find the number of years it takes for an initial investment of $500 to grow to$995 at an interest rate of 8% compounded continuously. We need to round the answer to the nearest whole number.

2. Solution Steps

The formula for continuous compounding is given by:

$A = Pe^{rt}$

where:

$A$ is the final amount

$P$ is the principal amount (initial investment)

$r$ is the interest rate (as a decimal)

$t$ is the time in years

In this problem, we have:

$A = 995$

$P = 500$

$r = 0.08$

We need to find $t$.

Plugging the given values into the formula, we get:

$995 = 500e^{0.08t}$

Divide both sides by 500:

$\frac{995}{500} = e^{0.08t}$

$1.99 = e^{0.08t}$

Take the natural logarithm of both sides:

$ln(1.99) = ln(e^{0.08t})$

$ln(1.99) = 0.08t$

Solve for $t$:

$t = \frac{ln(1.99)}{0.08}$

$t = \frac{0.688134}{0.08}$

$t = 8.601675$

Round the value of $t$ to the nearest whole number:

$t \approx 9$

3. Final Answer

9