Answer:
3.2 years.
Step-by-step explanation:
We are asked to find the number of years it will take an amount of $6,000 to be $8,000 compounded continuously at an annual rate of 9%.
We will use continuous compound interest formula to solve our given problem.
[tex]A=Pe^{rt}[/tex], where,
A = Amount after t years,
P = Principal amount,
e = Mathematical constant,
r = Interest rate in decimal form,
t = Time in years.
[tex]\$8,000=\$6,000e^{0.09*t}[/tex]
[tex]\frac{\$8,000}{\$6,000}=\frac{\$6,000e^{0.09*t}}{\$6,000}[/tex]
[tex]\frac{4}{3}=e^{0.09*t}[/tex]
Switch the sides:
[tex]e^{0.09*t}=\frac{4}{3}[/tex]
Now, we will take natural log of both sides.
[tex]\text{ln}(e^{0.09*t})=\text{ln}(\frac{4}{3})[/tex]
[tex]0.09*t=\text{ln}(\frac{4}{3})[/tex]
[tex]0.09*t=0.2876820724517808[/tex]
[tex]\frac{0.09*t}{0.09}=\frac{0.2876820724517808}{0.9}[/tex]
[tex]t=3.19646747\approx 3.2[/tex]
Therefore, it take approximately 3.2 years for the amount of $6,000 to be $8,000.
