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? - Imagemakers
Understanding Reservoir Water Levels: A Hydrology Model Analysis
Understanding Reservoir Water Levels: A Hydrology Model Analysis
When managing water resources during droughts, accurate modeling of reservoir levels is essential for sustainable planning. One such model used by hydrologists is the quadratic function:
$$
W(t) = -2t^2 + 12t + 50
$$
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.
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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
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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.
Conclusion
Using the hydrological model $ W(t) = -2t^2 + 12t + 50 $, the maximum water level reaches 68 meters at $ t = 3 $ months. This insight is valuable for drought preparedness and reservoir management, emphasizing the importance of timely data-driven decisions.
Keywords: hydrologist, water level model, reservoir modeling, drought impact, quadratic function, maximum water level, hydrology analysis
