Question: A sustainability project involves modeling energy output over time with the expression $ (3t - 2)(4t + 5) $. Expand this product and find the coefficient of the $ t $-term. - Sterling Industries
Understanding Energy Modeling in Sustainability: A Deep Dive
Understanding Energy Modeling in Sustainability: A Deep Dive
As climate action accelerates across the United States, modeling energy systems efficiently has become a cornerstone of strategic planning. One mathematical tool often behind the scenes supports these efforts: the expansion of quadratic expressions to simulate projected energy outputs over time. For those engaged in renewable infrastructure, policy modeling, or academic research, expressions like $ (3t - 2)(4t + 5) $ may appear when forecasting performance curves, cost-efficiency timelines, or environmental impact trends. Grasping how to expand such formulas goes beyond algebra—it unlocks insight into dynamic, real-world sustainability modeling. This article unpacks the expansion of $ (3t - 2)(4t + 5) $, focusing on identifying the coefficient of the $ t $-term, with relevance to current conversations about clean energy innovation and data-driven decision-making.
Understanding the Context
Why This Expression Trends in Sustainable Energy Modeling
The use of polynomial expressions like $ (3t - 2)(4t + 5) $ in sustainability projects reflects a growing need to represent changing variables over time. In modeling energy production, inputs such as sunlight exposure, wind consistency, or battery storage efficiency often shift and interact nonlinearly. Expressions combining linear terms — like $ at + b $ — can represent these interactive effects simply and powerfully. The $ t $-term specifically captures the linear, instantaneous change in output within a modeled timeframe, making it essential for early-stage assessments. As clean energy investments expand, accurately tracking these shifts becomes critical for forecasting performance, optimizing systems, and aligning policy with real-world variability.
How to Expand $ (3t - 2)(4t + 5) $: A Step-by-Step Breakdown
Key Insights
Expanding $ (3t - 2)(4t + 5) $ follows standard algebraic techniques and helps reveal the core dynamics of the system. Applying the distributive property (also called FOIL):
- First: $ 3t \cdot 4t = 12t^2 $
- Outer: $ 3t \cdot 5 = 15t $
- Inner: $ -2 \cdot 4t = -8t $
- Last: $ -2 \cdot 5 = -10 $
Combining all terms:
$$
(3t - 2)(4t + 5) = 12t^2 + 15t - 8t - 10
$$
Combine like terms:
$$
12t^2 + 7t - 10
$$
This expansion clearly identifies the coefficient of the $ t $-term as 7—a key insight when assessing timing and sensitivity in modeled outputs.
