We need to find the time (in years) it takes for an investment of $700 to earn $300 in interest, given an interest rate of 11.6% compounded continuously. We need to round the final answer to the nearest whole number.

1. Problem Description

We need to find the time (in years) it takes for an investment of $700 to earn$300 in interest, given an interest rate of 11.6% compounded continuously. We need to round the final answer to the nearest whole number.

2. Solution Steps

The formula for continuous compounding is:

$A = Pe^{rt}$

where:

$A$ is the final amount (principal + interest)

$P$ is the principal amount (initial investment)

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

$t$ is the time in years

In this problem:

$P = 700$

Interest earned $= 300$

Therefore, $A = P + \text{Interest} = 700 + 300 = 1000$

$r = 11.6\% = 0.116$

We need to find $t$.

Substitute the known values into the formula:

$1000 = 700e^{0.116t}$

Divide both sides by 700:

$\frac{1000}{700} = e^{0.116t}$

$\frac{10}{7} = e^{0.116t}$

Take the natural logarithm (ln) of both sides:

$\ln(\frac{10}{7}) = \ln(e^{0.116t})$

$\ln(\frac{10}{7}) = 0.116t$

Solve for $t$:

$t = \frac{\ln(\frac{10}{7})}{0.116}$

$t = \frac{\ln(10/7)}{0.116}$

$t \approx \frac{0.3566749439}{0.116}$

$t \approx 3.074783999$

Round the result to the nearest whole number:

$t \approx 3$

3. Final Answer

3