Respuesta :
Answer:
a) the amount of time in years it will take for the given investment to double is;
[tex]t=27.7\text{ years}[/tex]b) the amount of time in years it will take for the given investment to double is;
[tex]t=8.7\text{ years}[/tex]Explanation:
Given that under continuous compounding, the amount of time t in years required for an investment to double is a function of the interest rate r according to the formula;
[tex]t=\frac{\ln 2}{r}[/tex]a) we want to find the amount of time it will take a $3000 investment to reach $6000 (i.e double) for an interest rate of 2.5%.
[tex]r=2.5\text{\%=}\frac{\text{2.5}}{100}=0.025[/tex]Applying the given formula;
[tex]\begin{gathered} t=\frac{\ln 2}{0.025} \\ t=27.7\text{ years} \end{gathered}[/tex]Therefore, the amount of time in years it will take for the given investment to double is;
[tex]t=27.7\text{ years}[/tex]b) we want to find the amount of time it will take a $3000 investment to reach $6000 (i.e double) for an interest rate of 8%.
[tex]r=8\text{ \%}=\frac{8}{100}=0.08[/tex]Applying the given formula;
[tex]\begin{gathered} t=\frac{\ln 2}{0.08} \\ t=8.7\text{ years} \end{gathered}[/tex]Therefore, the amount of time in years it will take for the given investment to double is;
[tex]t=8.7\text{ years}[/tex]