Question

The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function L(t) = 0.01441t 3 − 0.4177t 2 + 2.703t + 1060.1 where t is measured in months since January 1, 2012. Estimate when the water level was highest during 2012.

Answer 1

Answer:

May 4th, 2012.

Step-by-step explanation:

The highest level can be found with the help of the First Derivative and Second Derivative Tests. First and second derivatives of the function are, respectively:

$l'(t) = 0.04323\cdot t^{2}-0.8354\cdot t +2.703$

$l''(t) = 0.08646\cdot t - 0.8354$

The First Derivative Tests consists on equalizing the first derivative to zero and finding the critical points.

$0.04323\cdot t^{2}-0.8354\cdot t +2.703 = 0$

Roots are $t_{1} \approx 15.672$ and $t_{2} \approx 4.077$. just the second root offer a realistic solution and is test by the second derivative.

$l''(4.077) = -0.482$ (which leads to a maximum).

Given that a year has 365 days or 12 months, the highest water level occurs at day:

$n = \frac{4.077}{12} \cdot (365\,days)$

$n \approx 124$ (May 4th).