Respuesta :
Answer:
37 years
Step-by-step Explanation:
When interest is being compounded continuously, we use the below compound interest formula;
[tex]A=Pe^{rt}[/tex]where A = future value of the investment
P = principal amount (initial investment)
r = interest rate in decimal
t = time in years
From the question, we're told that the future value(A) of the initial investment amount(P) will be triple, so we have that A = 3P.
Also, we're given an interest rate of 3%, so r = 3/100 = 0.03
Let's substitute these values into our formula;
[tex]3P=Pe^{0.03t}[/tex]We'll follow the below steps to solve for t;
Step 1: Divide both sides by P;
[tex]\begin{gathered} \frac{3P}{P}=\frac{Pe^{0.03t}}{P}^{} \\ 3=e^{0.03t} \end{gathered}[/tex]Step 2: We'll now take the natural logarithm of both sides;
[tex]\begin{gathered} \ln 3=\ln (e^{0.03t}) \\ \ln 3=0.03t\ln e \\ \text{Note that ln e = 1} \\ \ln 3=0.03t \end{gathered}[/tex]Step 3: Divide both sides by 0.03;
[tex]\begin{gathered} t=\frac{\ln 3}{0.03} \\ t=36.6 \\ t\approx37\text{ years} \end{gathered}[/tex]