The average residential electrical power use (in hundreds of...

The average residential electrical power use (in hundreds of watts) per hour is given in the following table. a. Compute the average total energy used in a day in kilowatt-hours (kWh). b. If a ton of coal generages 1842 $\mathrm{kWh}$ , how long does it take for an average residence to burn a ton of coal? c. Explain why the data might fit a plot of the form $p(t)=11.5-7.5 \sin \left(\frac{\pi t}{12}\right)$

The average residential electrical power use (in hundreds of watts) per hour is given in the following table.
a. Compute the average total energy used in a day in kilowatt-hours (kWh).
b. If a ton of coal generages 1842 $\mathrm{kWh}$ , how long does it take for an average residence to burn a ton of coal?
c. Explain why the data might fit a plot of the form $p(t)=11.5-7.5 \sin \left(\frac{\pi t}{12}\right)$

Key Concepts

Power and Energy Relationship

This concept explains the connection between power, which is the rate at which energy is used or generated, and energy, which is the total amount of work done or heat generated over time. Understanding that power (in watts or kilowatts) multiplied by time (in hours) gives energy (in watt-hours or kilowatt-hours) is fundamental in calculating daily energy consumption.

Unit Conversion in Energy Calculations

Accurate energy calculations depend on consistent units. This concept involves converting measurements such as watts to kilowatts or handling energy quantities in kilowatt-hours. Conversions allow one to align the units used in power ratings with those required for energy computations and comparisons, such as relating energy usage to fuel energy content.

Integration of Power over Time

When power consumption varies with time, the total energy usage is determined by integrating the power function over the given time period. In a discrete setting, this entails summing up the contributions from each interval (or hour) to find the average total energy used in a day. This concept is essential for aggregating variable power data into a cumulative energy metric.

Sinusoidal Models in Periodic Data Analysis

Many real-world phenomena, such as daily residential power use, exhibit periodic patterns that can be modeled using sinusoidal functions. This concept addresses why a sine or cosine function is appropriate for describing patterns that repeat cyclically. The periodic nature of human activity often results in predictable fluctuations in power usage, which sinusoidal models capture elegantly through parameters like amplitude, frequency, and phase shifts.

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