We can solve this problem using the formula for compound interest:
\[A = P\left(1 + \frac{r}{n}\right)^{n\cdot t}\]
Where:
- $A$ is the total amount of the investment at the end of the second month
- $P$ is the principal amount, which is $70
- $r$ is the annual interest rate, which is 12%
- $n$ is the number of times the interest is compounded per year, which is monthly and therefore 12
- $t$ is the total investment period in years, which is 2 months and therefore $\frac{2}{12}$
Plugging in the values, we have:
\[A = 70\left(1 + \frac{0.12}{12}\right)^{12\cdot \frac{2}{12}}\]
Simplifying within the parentheses:
\[A = 70\left(1 + \frac{0.01}{12}\right)^{2}\]
\[A = 70\left(1 + 0.000833\right)^{2}\]
And finally, calculating the power:
\[A = 70(1.000833)^{2}\]
\[A \approx 70(1.001666)\]
\[A \approx \boxed{70.12}\]