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

May 4th, 2012.

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 = (4.077)/(12) \cdot (365\,days)`

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