an initial investment of $200 is now valued at $350. the annual interest rate is 8% compounded continuously…

an initial investment of $200 is now valued at $350. the annual interest rate is 8% compounded continuously. the equation 200e^{0.08t}=350 represents the situation, where t is the number of years the money has been invested. about how long has the money been invested? use a calculator and round your answer to the nearest whole number. 5 years 7 years 19 years 22 years

an initial investment of $200 is now valued at $350. the annual interest rate is 8% compounded continuously. the equation 200e^{0.08t}=350 represents the situation, where t is the number of years the money has been invested. about how long has the money been invested? use a calculator and round your answer to the nearest whole number. 5 years 7 years 19 years 22 years

Answer

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation $200e^{0.08t}=350$ by 200. We get $e^{0.08t}=\frac{350}{200}=\frac{7}{4}$.

Step2: Apply natural - logarithm

Take the natural - logarithm of both sides. $\ln(e^{0.08t})=\ln(\frac{7}{4})$. Since $\ln(e^{x}) = x$, the left - hand side simplifies to $0.08t$. So, $0.08t=\ln(\frac{7}{4})$.

Step3: Solve for t

Divide both sides by 0.08. $t=\frac{\ln(\frac{7}{4})}{0.08}$. Using a calculator, $\ln(\frac{7}{4})\approx0.5596$ and $\frac{0.5596}{0.08}\approx6.995$. Rounding to the nearest whole number, $t\approx7$.

Answer:

7 years