APPLY IT Electricity Consumption The amount of electricity...

APPLY IT Electricity Consumption The amount of electricity (in trillion BTUs) consumed by U.S. residential customers in 2013 is given in the following table. Source: Energy Information Administration. (a) Plot the data, letting $t=1$ correspond to January, $t=2$ to February, and so on. Is it reasonable to assume that electri- cal consumption is periodic? (b) Use a calculator with trigonometric regression to find a trigonometric function of the form $$C(t)=a \sin (b t+c)+d$$ that models these data when $t$ is the month and $C(t)$ is the amount of electricity consumed (in trillion BTUs). Graph the function on the same calculator window as the data. (c) Determine the period, $T,$ of the function found in part (b). Discuss the reasonableness of this period. (d) Use the function from part (b) to estimate the consumption for the month of October, and compare it to the actual value.

APPLY IT Electricity Consumption The amount of electricity (in trillion BTUs) consumed by U.S. residential customers in 2013 is given in the following table. Source: Energy Information Administration.
(a) Plot the data, letting $t=1$ correspond to January, $t=2$ to February, and so on. Is it reasonable to assume that electri- cal consumption is periodic?
(b) Use a calculator with trigonometric regression to find a trigonometric function of the form
$$C(t)=a \sin (b t+c)+d$$
that models these data when $t$ is the month and $C(t)$ is the amount of electricity consumed (in trillion BTUs). Graph the function on the same calculator window as the data.
(c) Determine the period, $T,$ of the function found in part (b). Discuss the reasonableness of this period.
(d) Use the function from part (b) to estimate the consumption for the month of October, and compare it to the actual value.

Key Concepts

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. They are crucial in modeling phenomena that exhibit cyclic behavior, such as seasonal variations in electricity consumption, because they can capture the repetitive nature of the data over time.

Trigonometric Regression

Trigonometric regression is a statistical method used to model periodic data by fitting a trigonometric function, typically sine or cosine, to the observed data. It allows for the estimation of parameters that describe the amplitude, frequency, phase shift, and vertical shift, making it suitable for capturing the patterns of cyclic phenomena.

Sine Function Characteristics

The sine function is a basic trigonometric function that exhibits smooth, repetitive oscillations. By transforming this function using parameters for amplitude, phase shift, frequency, and vertical shift, one can model the behavior of natural phenomena that repeat over time, providing insights into how the variables change within each cycle.

Period Calculation from Function Parameters

The period of a sine function is determined by the frequency parameter and is calculated as the constant value that represents one complete cycle of the function. Understanding how to compute the period is essential in determining the cycle length in time series data, which helps in evaluating the regularity and reasonableness of the modeled behavior.

Data Modeling and Prediction

Data modeling using trigonometric functions involves fitting a model to historical data to capture the underlying patterns and subsequently using the model for prediction. This approach is valuable for making forecasts, comparing model outputs to actual measurements, and validating the suitability of the chosen mathematical model to the real-world data.

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