Answer
The highest water level in 2012occurs 4.1095 months, or $\approx$ 125 days after January 1, on May 6
Work Step by Step
To find the highest water level, we are looking for the maximum of the equation
$L(t)=0.01441t^{3}-0.4177t^{2}+2.703t+1060.1$
$L$ being the water level of Lake Lanier, $t$ being the time in months
In order to find the maximum, we need to find the derivative, and find when it's equal to zero
$L′(t)=0.04323t^{2}-0.8354t+2.703$
$L′(t)=0, t=4.1095, t=15.215$
Since we're only looking for the values of $L$ in 2012, the boundaries of $t$ are $0 \leq t \leq12$, so there's no need to check at $L(15.215)$
$L′(t) \gt 0$ when $t \lt 4.1095$
$L′(t) \lt 0$ when $ t \gt 4.1095$
The First Derivative Test gives us an absolute max when $t=4.1095$
In the year of 2012, the highest level of water occurs 4.1095 months, or $\approx$ 125 days into the year, or on May 6, 2012.
