Problem 60 - Precalculus with Calculus Previews

Question

Hours of Daylight The number \(H\) of daylight hours per day in various locations in the world can be modeled by a function of the form $$ H(t)=A \sin B(t-C)+D $$ where the variable \(t\) represents the number of days in a year corresponding to a specific calendar date (for example, February 1 corresponds to \(t=32\) In this problem we construct a model for Los Angeles, CA for the year 2017 (not a leap year) using data obtained from the U.S. Naval Observatory, Washington, D.C. (a) Find the amplitude \(A\) if 14.43 is the maximum number of daylight hours at the summer solstice and if 9.88 is the minimum number of daylight hours at the winter solstice. (b) Find \(B\) if the function \(H(t)\) is to have the period 365 days. (c) For Los Angeles in the year 2017 , we choose \(C\) \(=79 .\) Explain the significance of this number. [Hint: \(C\) has the same units as \(t\).] (d) Find \(D\) if the number of daylight hours at the vernal equinox for 2017 is 12.14 and occurs on March 20 (e) What does the model \(H(t)\) predict to be the number of daylight hours on January 1 ? On June 21 ? On August 1 ? On December 21 ? (f) Using a graphing utility to obtain the graph of \(H(t)\) on the interval [0,365] .

Step-by-Step Solution

Answer

The model parameters are \( A = 2.275 \), \( B = \frac{2\pi}{365} \), \( C = 79 \), \( D = 12.155 \), and they predict the daylight hours using the sine function. Graphing the model provides a visual representation of daylight variations over the year.

Step 1: Calculate Amplitude A

The amplitude of a sine function is given by the formula: \( A = \frac{\text{Maximum value} - \text{Minimum value}}{2} \). In this case, the maximum number of daylight hours is 14.43, and the minimum is 9.88. Thus, we calculate:\[ A = \frac{14.43 - 9.88}{2} = \frac{4.55}{2} = 2.275 \].

Step 2: Determine B Using Period

The period of a sine function is determined by the formula \( \frac{2\pi}{B} \). We know the function should have a period of 365 days, so we solve:\[ \frac{2\pi}{B} = 365 \]Solving for \( B \), we get:\[ B = \frac{2\pi}{365} \].

Step 3: Justify the Choice of C

The constant \( C \) is used to shift the sine wave horizontally. For Los Angeles, 2017, \( C = 79 \) represents the day of the year when the vernal equinox occurs, which is typically around March 20. This centers the wave such that the equinox aligns with the phase of the sine function at its average day length.

Step 4: Calculate the Vertical Shift D

The vertical shift \( D \) of the sine wave positions it to correspond with the average number of daylight hours at the equinox, which is given as 12.14 hours. The vertical shift is calculated as:\[ D = \frac{\text{Maximum value} + \text{Minimum value}}{2} = \frac{14.43 + 9.88}{2} = \frac{24.31}{2} = 12.155 \].

Step 5: Predict Daylight Hours on Specific Dates

Using the model \( H(t) = 2.275 \sin \left( \frac{2\pi}{365}(t-79) \right) + 12.155 \), compute the daylight hours for specific days: January 1 (\( t = 1 \)), June 21 (\( t = 172 \)), August 1 (\( t = 213 \)), and December 21 (\( t = 355 \)). Substitute each \( t \) into the equation to evaluate \( H(t) \).

Step 6: Graph H(t) Over One Year

Plot the function \( H(t) = 2.275 \sin \left( \frac{2\pi}{365}(t-79) \right) + 12.155 \) from \( t = 0 \) to \( t = 365 \) to visualize the daylight hours throughout the year. This graph can be obtained using a graphing utility.

Key Concepts

sine function

The sine function is a fundamental concept in trigonometry used to model periodic phenomena, like the hours of daylight over a year. At its core, the sine function, \(y = A \sin(Bx + C) + D\), represents a wave-like pattern.
The variables in this formula allow the function to be adjusted to fit real-world cycles, such as the length of daylight. Here, \(t\) stands for the day of the year, translating to varying positions on the graph. The value \(A\) is the amplitude, \(B\) determines the period, \(C\) adjusts the horizontal shift, and \(D\) signifies the vertical shift.
By modifying these parameters, one can accurately describe variations in natural events, with the sine function smoothly transitioning through its phases, similar to these events' cyclical nature.

amplitude in trigonometry

Amplitude in trigonometry represents the height of the wave from its mean position. In the context of daylight modeling, the amplitude (\(A\)) is calculated using the difference between the maximum and minimum values of daylight hours.
Amplitude can be seen as half the distance between the peak (maximum) and trough (minimum) of the wave. Mathematically, this is expressed as: \[ A = \frac{\text{Maximum value} - \text{Minimum value}}{2} \]For instance, with a maximum of 14.43 hours and a minimum of 9.88, the amplitude becomes 2.275.
This value provides insight into the range of daylight variability, symbolizing how much daylight oscillates around its average length over the year.

period of a function

The period of a function signals the duration it takes for the sine wave to complete one full cycle. In trigonometric modeling, it's crucial for showing recurring events, like the annual oscillation of daylight hours.
The formula for the period in a sine function is given by: \[ \text{Period} = \frac{2\pi}{B} \]If we want the model to have a period of 365 days, this sets the function to repeat yearly, matching the calendar's annual cycle. Solving for \(B\), we find: \(B = \frac{2\pi}{365}\).
Thus, the sine function realigns with its starting point every 365 days, making it ideal for representing yearly variation in daylight.

vertical shift in trigonometry

The vertical shift in trigonometry is the constant \(D\) in the sine function, shifting the entire graph up or down to align with the data’s average.
In daylight modeling, this vertical shift accounts for the mean daylight hours over the year, centering the wave around this average. The formula used is: \[ D = \frac{\text{Maximum value} + \text{Minimum value}}{2} \]With given values, it calculates to 12.155.
This shift ensures that the sine function oscillates around the average daylight length, accurately reflecting how daily daylight varies throughout different seasons, without altering the wave's inherent symmetry.