Question

The overhang of the roof of a house is designed to shade the windows for cooling in the summer and allow the Sun's rays to enter the house for heating in the winter. The Sun's angle of elevation, \(A,\) in degrees, at noon in Estevan, Saskatchewan, can be modelled by the formula \(A=-23.5 \sin \frac{360}{365}(x+102)+41\) where \(x\) is the number of days elapsed beginning with January 1. a) Use technology to sketch the graph showing the changes in the Sun's angle of elevation throughout the year. b) Determine the Sun's angle of elevation at noon on February 12. c) On what date is the angle of elevation the greatest in Estevan?

Short answer

The Sun's angle on February 12 is approximately 25 degrees. The angle is greatest on June 20.

Step-by-step solution

Understand the Given Function

Identify the components of the function provided: \(A=-23.5 \sin \left(\frac{360}{365}(x+102)\right)+41\). Here, \(A\) is the angle of elevation, and \(x\) represents the number of days since January 1st.

Sketch the Graph Using Technology

Use graphing software or a graphing calculator to plot the function. Input the equation \(A=-23.5 \sin \left(\frac{360}{365}(x+102)\right)+41\) and set the x-axis to represent the days of the year (from 0 to 365) and the y-axis to represent the angle of elevation.

Calculate the Sun's Angle on February 12

Convert February 12 into the number of days from January 1. It is the 43rd day of the year. Substitute \(x = 43\) into the equation.\[A = -23.5 \sin \left(\frac{360}{365}(43 + 102)\right) + 41\]Calculate the angle:

Simplify and Solve

First, simplify the expression inside the sine function:\[43 + 102 = 145\]. Then solve inside the sine function:\[\frac{360}{365} \cdot 145 \approx 142.74\].Now compute the sine value and substitute it in,\[A = -23.5 \sin(142.74) + 41\].Use a calculator to find \(\sin(142.74)\approx0.682\):\[A = -23.5 \cdot 0.682 + 41 \Leftrightarrow \-16.027+41 \Leftrightarrow A \approx 24.973\]

Find the Maximum Sun's Angle

Remember the maximum value of the sine function is 1. Therefore, \[A = -23.5 \sin(\theta) + 41\] is maximized when \(\sin(\theta) = -1\).\[A = -23.5 (-1) + 41 = 23.5 + 41\approx 64'5\].Solve for the value when \(\sin(\frac{360}{365}(x+102)) = -1\), which occurs at x corresponding to a date near June 20.Thus, the angle of elevation is at its greatest on the summer solstice, June 20.

Key concepts

Trigonometric Functions

Trigonometric functions are crucial in understanding the Sun's angle of elevation. They help describe the periodic behavior of natural phenomena, like the changing elevation of the Sun throughout the year. In this exercise, we use the sine function to model the Sun's elevation angle. The function provided is: $$A = -23.5 \sin\left(\frac{360}{365}(x+102)\right) + 41.$$Here, \(x\) represents the number of days since January 1, and \(A\) is the angle of elevation in degrees. The sine function is ideal for modeling periodic events due to its cyclical nature, with peaks and troughs that mirror the Sun’s changing position.

Graphing Technology

Graphing technology is essential for visualizing complex functions. To sketch the Sun's angle of elevation, you can use graphing software or a graphing calculator. Here’s a step-by-step guide:

• Input the equation \(A = -23.5 \sin\left(\frac{360}{365}(x+102)\right) + 41\).
• Set the x-axis to range from 0 to 365 to represent the days of the year.
• Set the y-axis to represent the angle of elevation, ranging approximately from the minimum to the maximum values of the equation.

Visualizing the graph helps you see the periodic nature of the angle of elevation, including the peaks representing the highest angles and the troughs representing the lowest angles. The graph provides a clear visual display of how the Sun's elevation changes through the year.

Seasonal Changes

Seasonal changes significantly affect the Sun's angle of elevation. As Earth orbits the Sun, different regions receive varying amounts of sunlight. This results in the changing angle of the Sun's elevation throughout the year. In Estevan, Saskatchewan, the provided function models these changes.On the summer solstice, around June 20, the Sun reaches its highest elevation. From the exercise, we calculated that the maximum angle of elevation is approximately 64.5°. Conversely, around the winter solstice, the Sun is at its lowest elevation. Understanding this model helps explain why areas experience more intense sunlight and longer days in summer while having shorter days and less sunlight in winter.