Problem 131 - Algebra and Trigonometry

Question

The number of hours of daylight in Boston is given by $$ y=3 \sin \left[\frac{2 \pi}{365}(x-79)\right]+12 $$where \(x\) is the number of days after January \(l\). Within a year, when does Boston have 10.5 hours of daylight? Give your answer in days after January 1 and round to the nearest day.

Step-by-Step Solution

Answer

Boston has approximately 10.5 daylight hours either on the 309th or the 50th day after January 1.

Step 1: Substitute the Given Value

First, set the function equal to 10.5 hours, which is the daylight hours given in the problem and substitute this into the original equation to solve for \(x\). This gives us: \[ 3 \sin\left[\frac{2\pi}{365}(x-79)\right] +12 = 10.5 \]

Step 2: Isolate the Sine Function

To begin isolating \(x\), subtract 12 from both sides of the equation and divide by 3. This yields: \[ \sin\left[\frac{2\pi}{365}(x-79)\right] = -0.5 \]

Step 3: Find the Argument of the Sine Function

To find \(x\), you need to solve the equation for \(x-79\). Remember that the sine function is negative in the third and fourth quadrants. Therefore, the argument of the sine function which is \(\frac{2\pi}{365}(x-79)\) is equal to either \(7\pi/6\) (in the third quadrant) or \(11\pi/6\) (in the fourth quadrant). This gives: \[ \frac{2\pi}{365}(x-79) = 7\pi/6, 11\pi/6 \]

Step 4: Solve for \(x\)

Lastly, solve for \(x\) by dividing by \((2\pi/365)\) and adding 79 to both sides. This gives: \[ x = 79+\frac{365}{2\pi}(7\pi/6), 79+\frac{365}{2\pi}(11\pi/6) \] After simplifying this equation, we come to a conclusion that \(x\) is approximately 309 or 50 days after January 1

Key Concepts

Daylight Hours

Daylight hours change throughout the year, affecting how long the sun stays above the horizon. This is influenced by Earth's tilt and orbit around the sun.
In this exercise, we look at how to determine specific days when daylight hours reach a particular length, like 10.5 hours. Understanding these variations is important for recognizing seasonal changes.

• Impact of Earth's Tilt: The tilt causes different daylight durations across seasons.
• Importance in Planning: Knowing daylight hours helps in planning outdoor activities.

This concept is linked to trigonometry as the yearly cycle can be modeled using trigonometric functions.

Sine Function

The sine function is key in modeling periodic phenomena like daylight hours. In this case, it captures the oscillation between long and short days.
The sine function, noted as \( \sin(x) \), oscillates between -1 and 1. It has properties that make it ideal for modeling the cyclical nature of daylight.

• Amplitude (3 in the equation): Indicates the height of the wave and affects how much the daylight hours vary.
• Phase Shift ((x-79) in the equation): Determines the horizontal shift, affecting when the cycle starts.

These parameters ensure the sine function accurately represents the daylight cycle over the year.

Equation Solving

Solving equations involving trigonometric functions involves isolating the function, identifying key angles, and using inverse trigonometric functions.
Here, the goal was to find when the daylight hours equal 10.5 by setting the function \( 3 \sin\left[\frac{2\pi}{365}(x-79)\right] +12 = 10.5 \) equal to 10.5.

• Step 1: Isolate Sine Function: Subtract 12 and divide by 3 to simplify.
• Step 2: Solve for Angles: Identify angles corresponding to the \sin\ value using known values like \frac{7\pi}{6} and \frac{11\pi}{6}.
• Step 3: Calculate Days: Solve for x to find specific days when this occurs.

Such problems often require a systematic approach to find precise solutions.

Periodicity in Trigonometry

Trigonometric functions exhibit periodicity, meaning they repeat at regular intervals. This property is crucial when modeling things like daylight hours that follow annual cycles.
The cycle or "period" for the sine function is evident as it repeats every \(2\pi\). In the equation given, the yearly cycle is reflected in the period \( \frac{2\pi}{365} \).

• Understanding Period: Defines how long before the function repeats.
• Relevance to Earth’s Orbit: Aligns with the Earth’s yearly orbit around the sun.

By leveraging periodicity, we can predict daylight duration at any given point in the year, showcasing the power of trigonometry in practical applications.