Respuesta :
The equation for the amount of money A in an account that earns interest compounded continuously, is:
[tex]A=A_0e^{rt}[/tex]Where A_0 represents the initial deposit, r is the interest rate and t represents time.
If the annual interest is 3%, then the interest rate r is:
[tex]r=\frac{3}{100}=0.03[/tex]If the amount of money triples after a time t, then the savings account will be worth $3000 at that time. Substitute A=3000, A_0=1000 and r=0.03:
[tex]3000=1000\cdot e^{0.03t}[/tex]Isolate t to find the amount of time needed for the investment to triple:
[tex]\begin{gathered} \Rightarrow\frac{3000}{1000}=e^{0.03t} \\ \Rightarrow3=e^{0.03t} \end{gathered}[/tex]Take the natural logarithm to both sides of the equation:
[tex]\ln (3)=\ln (e^{0.03t})[/tex]The natural logarithm of e^0.03t is equal to 0.03t. Then:
[tex]\begin{gathered} 0.03t=\ln (3) \\ \Rightarrow t=\frac{\ln (3)}{0.03} \end{gathered}[/tex]Use a calculator to evaluate the expression:
[tex]\begin{gathered} \Rightarrow t=\frac{1.0986\ldots}{0.03} \\ \Rightarrow t=36.62\ldots \end{gathered}[/tex]Therefore, to the nearest hundredth, the amount of time needed for the investment to triple, is:
[tex]36.62\text{ years}[/tex]