Question

an initial investment of $100 is now valued at $150. the annual interest rate is 5%, compounded continuously. the equation $100e^{0.05t}=150$ represents the situation, where $t$ is the number of years the money has been invested. about how long has the money been invested? use your calculator and round to the nearest whole number. years

Explanation:

Step1: Divide both sides by 100

$$\frac{100e^{0.05t}}{100}=\frac{150}{100}$$
$$e^{0.05t}=1.5$$

Step2: Take the natural logarithm of both sides

$$\ln(e^{0.05t})=\ln(1.5)$$
Using the property $\ln(e^{x}) = x$, we get $0.05t=\ln(1.5)$

Step3: Solve for $t$

$$t=\frac{\ln(1.5)}{0.05}$$
Using a calculator, $\ln(1.5)\approx0.4055$, then $t=\frac{0.4055}{0.05}=8.11$

Answer:

$8$