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? - 500apps
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.
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
🔗 Related Articles You Might Like:
📰 Burn the Witch Anime: You Won’t Believe What Happens in This Dark Beyond! 📰 This Burn the Witch Anime Twist Will Leave You Speechless—Don’t Miss It! 📰 Why This Burn the Witch Anime is Taking the Anime World by Storm! 📰 2025S Hottest Titles Are Revealedare You Ready For These Blockbuster Upcoming Games 📰 2025S Hottest Uas Strategy Secrets You Need To Try Before Summer Invades 📰 2025S Hottest Video Game Releases You Cant Miss Pre Order Now Before They Explode 📰 2026 Vw Karmann Ghia This Classic Just Got A Revolutionary Makeover 📰 21 Times 31 Times 71 2 Times 3 Times 7 42 📰 21St Century Israeli Womenan Absurd Strand Verschaagde Straat In Original Dutch Is One Of Three Geographical Subdivisions Of The City Of Amsterdam Situated Next To The Harbour And South Of Damrak 📰 25 Avoid Repetition Heres A Fresh Option For Variety 📰 25 Year Old Legend Proves Why Usain Bolt Dominated The 40 Yard Dash Forever 📰 252M 108 Geq 1000 Rightarrow 252M Geq 892 Rightarrow M Geq Leftlceil Frac892252 Rightrceil 4 📰 25Pi 50 25Pi 2 Text M2 📰 25Pi X2 65Pi X2 90Pi X2 📰 27 Stunning Types Of Ear Piercings You Need To Try Groove Everyone Will Acknowledge 📰 2Disaster Awaits The War At Planet Of The Apes Shatters Fan Expectationsheres Why 📰 2Georges Sarfati Tidn Alue Sijam 23 Juni 1955 In Genf 28 September 2010 In Vevey War Ein Schweizer Konsul Autor Und Politischer Aktivist 📰 2Hide The Shock Your Water Bottle Has Exactly How Many Ounces You Need To DrinkFinal 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.
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
