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 generates 1842 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 generates 1842 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

Trigonometric Modeling of Periodic Behavior

Trigonometric functions, especially sine and cosine curves, are often used to model periodic phenomena because of their inherent periodicity. In this context, a function of the form p(t) = 11.5 - 7.5 sin(?t/12) is used to represent the cyclical nature of residential electrical consumption over the course of a day. The amplitude represents the extent of variation from the average consumption, while the phase and period are aligned with the daily cycle, offering a concise mathematical model of the fluctuations observed in residential energy use.

Energy Equivalence with Resource Consumption

This concept pertains to comparing the energy consumption of a residence to the energy output of a resource, such as a ton of coal. By knowing the energy yield of a fuel source (e.g., 1842 kWh per ton of coal), one can estimate how long it takes for an average residence's energy use to equal the energy generated by that fuel amount. This kind of computation is useful in discussions on energy efficiency, sustainability, and environmental impact.

Electrical Energy Consumption

This concept involves calculating the total amount of electrical energy used over a period of time based on the power consumption rates. It requires understanding that power (normally measured in watts or kilowatts) multiplied by time (in hours) gives energy (in watt-hours or kilowatt-hours). This relationship is fundamental when assessing daily or periodic energy usage in residential contexts.

Unit Conversion in Energy Calculations

Understanding unit conversion is crucial when dealing with power and energy calculations. In many cases, the power values might be given in units such as hundreds of watts, which need to be converted to kilowatts before multiplying by time to find energy in kilowatt-hours. Accurate unit conversion ensures that all quantities are consistent and that the final energy usage is computed correctly.

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