Problem 17 - Applied Calculus

Question

The number of hours, \(H\), of daylight in Madrid as a function of date is approximated by the formula $$ H=12+2.4 \sin [0.0172(t-80)] $$ where \(t\) is the number of days since the start of the year. Find the average number of hours of daylight in Madrid: (a) in January (b) in June (c) over a year (d) Explain why the relative magnitudes of your answers to parts (a), (b), and (c) are reasonable.

Step-by-Step Solution

Answer

(a) ~10 hours; (b) ~14.4 hours; (c) ~12 hours; (d) Seasonal variations explain differences.

Step 1: Understand the formula basics

The formula, \( H = 12 + 2.4 \sin [0.0172(t-80)] \), represents the number of daylight hours as a sinusoidal function of the day \( t \). The sine function fluctuates between -1 and 1, causing the daylight hours to oscillate around 12, with an amplitude of 2.4 hours.

Step 2: Calculate average daylight in January

January covers the first 31 days (\( t = 1 \) to \( t = 31 \)). We need to evaluate and average \( H \) over these days. This involves calculating \( H \) for each day and finding the mean. However, due to symmetry in trigonometric functions over intervals, a rough estimate uses the midpoint: \( t = 16 \). Substitute \( t = 16 \) into the formula: \[ H = 12 + 2.4 \sin[0.0172(16-80)] \approx 12 + 2.4 \sin(-1.1032) \approx 12 - 2.01 \approx 9.99 \] The average is approximately 10 hours.

Step 3: Calculate average daylight in June

June covers \( t = 152 \) to \( t = 181 \). Use the midpoint \( t = 166.5 \). Substitute \( t = 166.5 \) in the formula: \[ H = 12 + 2.4 \sin[0.0172(166.5-80)] \approx 12 + 2.4 \sin(1.48666) \approx 12 + 2.4 \times 0.996 \approx 14.39 \] Thus, the average is approximately 14.4 hours.

Step 4: Calculate average daylight over a year

Over a year, \( t = 1 \) to \( t = 365 \). The average value of the sine function over its entire period is zero. Hence, the yearly average daylight is: \[ H = 12 + 2.4 \times 0 = 12 \] The average daylight over the entire year is 12 hours.

Step 5: Reasoning behind magnitudes

- In January, daylight hours are fewer due to the winter solstice, which explains the lower average compared to the year-long average. - In June, close to the summer solstice, daylight hours peak, hence the higher average. - The overall yearly average is 12 hours, as expected, due to the symmetry of the sine function around the zero axis, causing long summer days and short winter days to balance each other out.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical tools that help describe periodic phenomena. They include sine, cosine, and tangent functions. Their key characteristic is their ability to model waves and oscillations.

In daylight calculation, the sine function is particularly useful. The sine function oscillates between -1 and 1, creating a wave-like pattern. This is perfect for modeling things that repeat over time, like daylight hours throughout a year.

For the formula: \[ H = 12 + 2.4 \sin [0.0172(t-80)] \],
the sine component \( \sin [0.0172(t-80)] \) allows the calculation of oscillations. Each point on the graph of a sine wave represents a time of day and shows how many hours of daylight there are. Using trigonometric functions simplifies analyzing and predicting cyclical behaviors like daylight changes.

Sinusoidal Models

A sinusoidal model is a specific type of mathematical expression used to represent periodic cycles that repeat at regular intervals, like daylight hours.

These models use the sine function to describe how some quantity varies according to time. In our formula:
\[ H = 12 + 2.4 \sin [0.0172(t-80)] \],
the sinusoidal model captures the essence of daylight changes over the year.

• The constant 12 indicates the midpoint or average daylight hours without any variation.
• The number 2.4 refers to the amplitude. It shows the maximum deviation from the average, which means the daylight can increase or decrease by up to 2.4 hours from 12.
• The expression \( 0.0172(t-80) \) determines the frequency and phase shift—it compresses and shifts the function to match the actual passage of days.

Sinusoidal models help engineers, scientists, and students understand and predict systems with natural cyclical behavior.

Daylight Hours Analysis

Analyzing daylight hours using mathematical formulas involves understanding how daylight changes throughout the year based on earth's tilt and orbit around the sun. The formula provided for Madrid helps estimate these changes systematically.

For different times of the year:

• **Winter months**: In January, fewer daylight hours occur due to the position of the Earth, aligned such that the Northern Hemisphere receives less sunlight. The analysis showed that the average daylight is about 10 hours.
• **Summer months**: During June, when the Northern Hemisphere tilts towards the sun, days become longer, leading to an average of around 14.4 daylight hours.
• **Yearly average**: When averaging daylight over the whole year, the values of cosine and sine over their period generally balance out to zero, leading us to an average of 12 hours. This balancing act between longer and shorter days throughout the year is typical for sinusoidal behavior, effectively showcasing how all highs and lows neutralize over the long term.

Daylight hours analysis aids in grasping how natural cycles work, and these calculations are crucial for fields like agriculture, science, and education, where understanding sunlight's impact is vital.