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:
[tex]l'(t) = 0.04323\cdot t^{2}-0.8354\cdot t +2.703[/tex]
[tex]l''(t) = 0.08646\cdot t - 0.8354[/tex]
The First Derivative Tests consists on equalizing the first derivative to zero and finding the critical points.
[tex]0.04323\cdot t^{2}-0.8354\cdot t +2.703 = 0[/tex]
Roots are [tex]t_{1} \approx 15.672[/tex] and [tex]t_{2} \approx 4.077[/tex]. just the second root offer a realistic solution and is test by the second derivative.
[tex]l''(4.077) = -0.482[/tex] (which leads to a maximum).
Given that a year has 365 days or 12 months, the highest water level occurs at day:
[tex]n = \frac{4.077}{12} \cdot (365\,days)[/tex]
[tex]n \approx 124[/tex] (May 4th).
