Question: A hydrologist models the water level in a reservoir as $ W(t) = -2t^2 + 12t + 50 $, where $ t $ is time in months since the start of the drought. What is the maximum water level, and at what time does it occur? - Appfinity Technologies

Understanding the Context

where $ W(t) $ represents the water level in meters and $ t $ is the time in months since the onset of a drought. This article explores the key features of the model—specifically, identifying the maximum water level and the time at which it occurs.

Finding the Maximum Water Level

The given equation is a quadratic function in standard form $ W(t) = at^2 + bt + c $, with $ a = -2 $, $ b = 12 $, and $ c = 50 $. Since the coefficient of $ t^2 $ is negative ($ a < 0 $), the parabola opens downward, meaning it has a single maximum point at its vertex.

Key Insights

The time $ t $ at which the maximum occurs is given by the vertex formula:

$$
t = -rac{b}{2a} = -rac{12}{2(-2)} = -rac{12}{-4} = 3
$$

So, the maximum water level happens 3 months after the drought begins.

Calculating the Maximum Water Level

Final Thoughts

To find the actual maximum water level, substitute $ t = 3 $ back into the original equation:

$$
W(3) = -2(3)^2 + 12(3) + 50 = -2(9) + 36 + 50 = -18 + 36 + 50 = 68
$$

Thus, the maximum water level in the reservoir is 68 meters.

Interpretation and Drought Implications

This model illustrates that water levels peak early in a drought (after 3 months) before beginning a steady decline. Understanding this pattern helps water managers plan conservation measures, allocate supplies, and prepare for scarcity. The peak at $ t = 3 $ months with a level of 68 meters highlights a critical window for intervention—before levels drop rapidly.