Answer:
It will take 22 years before the investment triples.
Step-by-step explanation:
To determine how many years it will take for an investment to triple if interest if compounded continuously at 5%, use the continuous compounding interest formula.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]
Given values:
- A = 3P (triple the principal amount)
- P = P
- r = 5% = 0.05
Substitute the given values into the formula and solve for t.
[tex]\implies 3P=Pe^{0.05t}[/tex]
Divide both sides of the equation by P:
[tex]\implies 3=e^{0.05t}[/tex]
Take natural logs of both sides of the equation:
[tex]\implies \ln 3=\ln e^{0.05t}[/tex]
[tex]\textsf{Apply the log power law:} \quad \ln x^n=n \ln x[/tex]
[tex]\implies \ln 3=0.05t\:\ln e[/tex]
As ln e = 1, then:
[tex]\implies \ln 3=0.05t[/tex]
Divide both sides of the equation by 0.05:
[tex]\implies \dfrac{\ln 3}{0.05}=t[/tex]
[tex]\implies t=20\ln3[/tex]
[tex]\implies t=21.9722457...[/tex]
[tex]\implies t=22\;\sf years\;(nearest\;year)[/tex]
Therefore, it will take 22 years before the investment triples.
