Daylight Hours in New Orleans The seasonal variation in...

Daylight Hours in New Orleans The seasonal variation in length of daylight can be modeled by a sine function. For example, the daily number of hours of daylight in New Orleans is given by $$ h=\frac{35}{3}+\frac{7}{3} \sin \frac{2 \pi x}{365} $$ where $x$ is the number of days after March 21 (disregarding leap year). (Source: Bushaw, D., et al., $A$ Sourcebook of Applications of School Mathematics, Mathematical Association of America.) (a) On what date will there be about 14 hr of daylight? (b) What date has the least number of hours of daylight? (c) When will there be about 10 hr of daylight?

Daylight Hours in New Orleans The seasonal variation in length of daylight can be modeled by a sine function. For example, the daily number of hours of daylight
in New Orleans is given by
$$
h=\frac{35}{3}+\frac{7}{3} \sin \frac{2 \pi x}{365}
$$
where $x$ is the number of days after March 21 (disregarding leap year). (Source:
Bushaw, D., et al., $A$ Sourcebook of Applications of School Mathematics, Mathematical Association of America.)
(a) On what date will there be about 14 hr of daylight?
(b) What date has the least number of hours of daylight?
(c) When will there be about 10 hr of daylight?

Key Concepts

Interpreting Variables in Context

Understanding the role of variables in the model is key, where one variable represents time (days) and the other represents a physical measurement (daylight hours). This interpretation allows for the conversion of mathematical results into meaningful real-life dates or events.

Solving Trigonometric Equations

This concept involves manipulating trigonometric equations to determine the input values (such as days) that produce specific output values (such as a certain number of daylight hours). It is essential for finding dates corresponding to particular scenarios in periodic models.

Phase Shift and Horizontal Translation

The phase shift of a sine function accounts for any horizontal translation of the graph. This is key when the starting point of the cycle does not align with the origin of the sine function, such as when the cycle begins on a date other than January 1, necessitating an adjustment to correctly model the timing of the peaks and troughs.

Amplitude and Vertical Shift

In the context of a sine function, the amplitude represents the extent of variation from the central value, while the vertical shift (or midline) adjusts the entire function up or down. Together, these parameters define the maximum and minimum values of the modeled phenomenon, such as the range of daylight hours.

Sine Function Modeling

The sine function is commonly used to model periodic phenomena, such as the variation in daylight hours over the course of a year. This concept involves understanding how a sine function can represent regular, repeating patterns, where the values oscillate between a maximum and a minimum in a predictable way.

Period of a Sine Function

The period of a sine function is the length of the interval required for the function to complete one full cycle. This is determined by the coefficient inside the sine function and is crucial for accurately modeling phenomena that repeat cyclically, such as the annual cycle of daylight changes.

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