An electrical engineer is modeling power output over time. A solar array produces 400 kW in peak sunlight, but output decreases quadratically due to temperature, modeled by P(t) = 400 − 10t², where t is hours after noon. What is the total energy generated from noon to 4 PM?

["Modeling Solar Power Output: Total Energy Generated from Noon to 4 PM", "Understanding how solar arrays produce electricity throughout the day is essential for optimizing energy grids, storage systems, and power forecasts. In real-world conditions, solar power output doesn’t remain constant—especially under fluctuating temperatures and solar intensity. For a solar array with peak output of 400 kW at noon, one effective model uses a quadratic function to represent how power decreases over time:", "[ P(t) = 400 - 10t^2 ]", "where ( t ) denotes the number of hours after noon. This model assumes temperature effects reduce output quadratically as solar irradiance dips and panel temperatures rise.", "Calculating Total Energy Over Time", "Energy generation is the integral of power over time. To find the total energy produced from noon (( t = 0 )) to 4 PM (( t = 4 )), compute the definite integral of ( P(t) ) from 0 to 4:", "[\n\ ext{Energy} = \int_0^4 P(t), dt = \int_0^4 \left(400 - 10t^2\right) dt\n]", "We integrate term by term:", "[\n\int_0^4 400, dt = 400t \Big|_0^4 = 400 \ imes 4 = 1600\n]", "[\n\int_0^4 10t^2, dt = 10 \cdot \frac{t^3}{3} \Big|_0^4 = 10 \cdot \left(\frac{64}{3}\right) = \frac{640}{3} \approx 213.33\n]", "Subtracting the two results:", "[\n\ ext{Energy} = 1600 - \frac{640}{3} = \frac{4800 - 640}{3} = \frac{4160}{3} \approx 1386.67 \ ext{ kWh}\n]", "Conclusion", "The solar array produces a total of approximately 1,386.67 kWh of energy from noon to 4 PM, modeling the quadratic decline in output due to environmental factors. This calculation helps engineers forecast renewable energy contributions and improve grid integration strategies. By leveraging precise mathematical modeling, electrical engineers can better predict performance, plan storage, and maximize the efficiency of solar power systems.", "For accurate energy modeling, real-world data often refines such quadratic approximations, incorporating variable sun angles, cloud cover, and cooling effects—but the core principle remains: understanding output decay via functions like ( P(t) ) is key to optimizing renewable energy systems."]