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.\no 5 years\no 7 years\no 19 years\no 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.\no 5 years\no 7 years\no 19 years\no 22 years

Answer

Answer:

B. 7 years

Explanation:

Step1: Isolate the exponential term

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

Step2: Take natural - logarithm of both sides

$\ln(e^{0.08t})=\ln(\frac{7}{4})$.

Step3: Use the property $\ln(e^x)=x$

$0.08t = \ln(\frac{7}{4})$.

Step4: Solve for $t$

$t=\frac{\ln(\frac{7}{4})}{0.08}$. Since $\ln(\frac{7}{4})\approx\ln(1.75)\approx0.5596$, then $t=\frac{0.5596}{0.08}\approx6.995\approx7$ years.