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.\nyears\ndone
Answer
Explanation:
Step1: Isolate the exponential term
Divide both sides of the equation $100e^{0.05t}=150$ by 100. $e^{0.05t}=\frac{150}{100}=1.5$
Step2: Take the natural - logarithm of both sides
Since $\ln(e^{x}) = x$, taking the natural logarithm of both sides of $e^{0.05t}=1.5$ gives $\ln(e^{0.05t})=\ln(1.5)$. $0.05t=\ln(1.5)$
Step3: Solve for $t$
Divide both sides of the equation $0.05t=\ln(1.5)$ by 0.05. $t=\frac{\ln(1.5)}{0.05}$ Using a calculator, $\ln(1.5)\approx0.4055$ and $\frac{0.4055}{0.05}=8.11$.
Answer:
8