an initial investment of $100 is now valued at $150. the annual interest rate is 5%, compounded…
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.
Answer
Explanation:
Step1: Isolate the exponential term
Divide both sides of the equation $100e^{0.05t}=150$ by 100. We get $e^{0.05t}=\frac{150}{100} = 1.5$.
Step2: Take the natural - logarithm of both sides
Using the property $\ln(e^{x})=x$, we have $\ln(e^{0.05t})=\ln(1.5)$. So, $0.05t=\ln(1.5)$.
Step3: Solve for $t$
Divide both sides of the equation $0.05t=\ln(1.5)$ by 0.05. Then $t=\frac{\ln(1.5)}{0.05}$. Using a calculator, $\ln(1.5)\approx0.4055$ and $\frac{0.4055}{0.05}=8.11$. Rounding to the nearest whole number, $t\approx8$.
Answer:
8