Answer:
About 27 years.
Step-by-step explanation:
Byron deposits $13,500 in an account that earns 2.6% interest compounded monthly, and we want to determine how many years it will take for Byron's money to double.
Recall that compound interest is given by the formula:
[tex]\displaystyle A = P\left( 1 + \frac{r}{n}\right)^{nt}[/tex]
Since our initial deposit is $13,500 at a rate of 2.6% compounded monthly, P = 13500, r = 0.026, and n = 12:
[tex]\displaystyle \begin{aligned} A &= (13500)\left( 1 + \frac{(0.026)}{(12)}\right)^{(12)t} \\ \\ &=13500\left(\frac{6013}{6000}\right)^{12t} \end{aligned}[/tex]
For his deposit to double, A must equal $27,000. Hence:
[tex]\displaystyle (27000) &=13500\left(\frac{6013}{6000}\right)^{12t}[/tex]
Solve for t:
[tex]\displaystyle \begin{aligned} 27000 &= 13500 \left(\frac{6013}{6000}\right)^{12t} \\ \\ \left(\frac{6013}{6000}\right)^{12t} &= 2 \\ \\ \ln\left(\frac{6013}{6000}\right)^{12t} &= \ln (2)\\ \\ 12t \ln \frac{6013}{6000} &= \ln 2 \\ \\ t &= \frac{\ln 2}{12\ln \dfrac{6013}{6000}} = 26.6883... \approx 27\end{aligned}[/tex]
In conclusion, it will take Byron about 27 years for his deposit to double.
