Answer
The water level was the highest 4.11 months after January 1, which was on about May 4th, 2012.
Work Step by Step
We can find the value of $t$ when $L'(t) = 0$:
$L(t) = 0.01441t^3- 0.4177t^2 + 2.703t + 1060.1$
$L'(t) = 0.04323t^2- 0.8354t + 2.703 = 0$
We can use the quadratic formula to find the values of $t$:
$t = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
$t = \frac{-(- 0.8354) \pm \sqrt{(-0.8354)^2-4(0.04323)( 2.703)}}{(2)(0.04323)}$
$t = \frac{0.8354 \pm \sqrt{(-0.8354)^2-4(0.04323)( 2.703)}}{0.08646}$
$t = 4.11~~$ or $~~t = 15.22$
Since the domain is between $0$ and $12$, the solution is $t = 4.11$
We can verify the water level for $t = 4.11$ and the endpoints of the interval $t = 0$ and $t = 12$:
$L(4.11) = 0.01441(4.11)^3- 0.4177(4.11)^2 + 2.703(4.11) + 1060.1 = 1065.2$
$L(0) = 0.01441(0)^3- 0.4177(0)^2 + 2.703(0) + 1060.1 = 1060.1$
$L(12) = 0.01441(12)^3- 0.4177(12)^2 + 2.703(12) + 1060.1 = 1057.3$
During 2012, the water level reached a highest level of 1065.2 feet above mean sea level. This occurred 4.11 months after January 1.
The water level was the highest 4.11 months after January 1, which was on about May 4th, 2012.
