The seasonal variation in length of daylight can be modeled...

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 the equation $$ h=\frac{35}{3}+\frac{7}{3} \sin \frac{2 \pi x}{365} $$ where $x$ is the number of days after March 21 (disregarding leap year). (Data from Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, National Council of Teachers of Mathematics.) (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?

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 the equation
$$
h=\frac{35}{3}+\frac{7}{3} \sin \frac{2 \pi x}{365}
$$
where $x$ is the number of days after March 21 (disregarding leap year). (Data from Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, National Council of Teachers of Mathematics.)
(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

Sine Functions as Models for Periodic Phenomena

Sine functions are ideal for modeling events that repeat in a regular, cyclical pattern, such as the changing length of daylight over the year. Their periodicity allows them to capture the rise and fall seen in many natural phenomena, making them a fundamental tool in the study of oscillatory behavior and seasonality.

Transformation of Trigonometric Functions

Understanding the transformations applied to a basic sine function—such as vertical shifts, amplitude adjustments, and changes in period—is crucial for accurately modeling real-world data. In the given equation, the vertical shift adjusts the average daylight hours, the amplitude represents the extent of variation from this average, and the period reflects the length of the complete seasonal cycle.

Solving Trigonometric Equations

To determine specific values or dates from a trigonometric model, one must solve equations involving the sine function. This often involves isolating the sine term, applying inverse trigonometric functions, and considering the periodicity of sine to find all possible solutions in the context of the problem.

Mapping Mathematical Results to Real-World Contexts

Interpreting the solution of a trigonometric equation requires converting abstract mathematical outcomes into tangible real-world dates or times. This involves understanding the domain of the function—in this case, days after a reference date—and accurately translating mathematical solutions into a calendar or timeline.

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