Question

Verify that the function $$ D(t)=2.8 \sin \left(\frac{2 \pi}{365}(t-81)\right)+12 $$ has the following properties, where \(t\) is measured in days and \(D\) is measured in hours. a. It has a period of 365 days. b. Its maximum and minimum values are 14.8 and \(9.2,\) respectively, which occur approximately at \(t=172\) and \(t=355\) respectively (corresponding to the solstices). c. \(\overline{D(81)}=12\) and \(D(264)=12\) (corresponding to the equinoxes).

Short answer

Question: Verify the properties of the function $$D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12$$, where D(t) represents the duration of daylight in hours and t is the number of days since the beginning of the year. Answer: The function D(t) is periodic with a period of 365 days. Its maximum value is 14.8 hours, occurring on the 172nd day of the year, and its minimum value is 9.2 hours, occurring on the 355th day of the year. The function has a value of 12 hours at both t = 81 and t = 264, corresponding to the equinoxes.

Step-by-step solution

Understand the given function

The function given is: $$ D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12. $$ This is a sinusoidal function with amplitude 2.8, period 365, a horizontal shift of 81 days, and 12 hours of average daylight duration.

Verify the period

To verify the period, we will check that the function D(t) is periodic with a period of 365 days. To be periodic, the following must hold true: $$ D(t + n \cdot \textrm{period}) = D(t) $$ For any integer n. We will substitute t + 365 for t: $$ D(t + 365) = 2.8 \sin\left(\frac{2 \pi}{365}(t + 365 - 81)\right) + 12 = 2.8 \sin\left(\frac{2 \pi}{365}(t - 81) + 2 \pi\right) + 12 $$ As the sine function has a period of \(2\pi\), we have that: $$ D(t + 365) = 2.8 \sin\left(\frac{2 \pi}{365}(t - 81)\right) + 12 = D(t), $$ which shows that the function is indeed periodic with a period of 365 days.

Verify maximum and minimum values

To verify the maximum and minimum values of the function, we can find the critical points, where the derivative of the function is zero. The derivative of the function is: $$ D'(t) = 2.8 \cdot \frac{2 \pi}{365} \cos\left(\frac{2 \pi}{365}(t-81)\right) $$ The critical points occur when \(D'(t) = 0\), i.e., when the cosine part is zero: $$ \cos\left(\frac{2 \pi}{365}(t-81)\right) = 0 $$ From the cosine function, we know that this occurs at: $$ \frac{2 \pi}{365}(t-81) = (2n+1)\frac{\pi}{2}, \quad n \in \mathbb{Z} $$ We will solve for t: $$ t - 81 = \frac{365}{2 \pi}(2n + 1)\frac{\pi}{2} \Rightarrow t = \frac{365}{2}(2n+1) + 81 $$ For n = 0 and n = 1, we get t = 172 and t = 355, respectively, which correspond to the function's maximum and minimum values. From the given function, we can find the maximum and minimum values: $$ D(172) = 2.8 \sin\left(\frac{2 \pi}{365}(172 - 81)\right) + 12 = 14.8 $$ and $$ D(355) = 2.8 \sin\left(\frac{2 \pi}{365}(355 - 81)\right) + 12 = 9.2 $$ These values confirm that the maximum and minimum values are 14.8 and 9.2, respectively.

Verify the stated properties at t = 81 and t = 264

We must check whether the given function has the properties mentioned about the equinoxes: $$ D(81) = 2.8 \sin\left(\frac{2 \pi}{365}(81 - 81)\right) + 12 = 2.8 \sin(0) + 12 = 12 $$ and $$ D(264) = 2.8 \sin\left(\frac{2 \pi}{365}(264 - 81)\right) + 12 = 2.8 \sin\left(\pi\right) + 12 = 12 $$ This verifies that the function has the specified properties at t = 81 and t = 264 (the equinoxes).

Key concepts

Sinusoidal Functions

Sinusoidal functions are a type of mathematical function that describe a smooth, repetitive oscillation. These functions are closely related to the sine wave, a fundamental component in the study of trigonometry. Sinusoidal functions are commonly used to model periodic phenomena found in nature, such as sound waves, light waves, and the alternating current in electrical systems.
The general form of a sinusoidal function can be expressed as:

• \(f(t) = A \, ext{sin}(Bt + C) + D\)

where:

• \(A\) represents the amplitude, which is the peak value of the function.
• \(B\) determines the frequency, influencing how often the wave repeats.
• \(C\) is the phase shift, which translates the wave horizontally.
• \(D\) shifts the wave vertically and is often referred to as the vertical shift or midline.

In our function, \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\), we see that it closely fits the sinusoidal model, highlighting its role in modeling real-world periodic events, such as daylight duration in different months of the year.

Periodicity in Functions

Periodicity refers to the property of a function that repeats its values at regular intervals or periods. An important characteristic in the study of trigonometric functions, periodicity is a fundamental concept in understanding the cyclic nature of sinusoidal functions.
The period of a function is the length of the smallest interval over which the function completes one full cycle. For example, the sine function, \(\sin(x)\), has a period of \(2\pi\), meaning the function repeats itself every \(2\pi\) units along the x-axis.
For the function \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\), the period is clearly set by the term \(\frac{2\pi}{365}\). This signifies that the function repeats itself every 365 units, attributed to the annual cycle of season changes over a year.

Amplitude in Trigonometric Functions

The amplitude of a trigonometric function measures the distance from the midline to the peak values. It quantifies how much the function above and below its average value oscillates. The amplitude is always measured as a positive number. This concept is crucial for understanding how "tall" or "short" the function's oscillations are in a physical sense.
For the sine function, the amplitude is the coefficient of the sine, given as \(A\) in the general formula \(f(t) = A \sin(Bt + C) + D\).
In our function example, \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\), the amplitude is \(2.8\). This signifies that the maximum deviation of \(D(t)\) from its mean value of 12 is 2.8 hours, resulting in maximum and minimum daylight values of 14.8 and 9.2 hours, respectively.

Calculus Derivatives

Calculus derivatives are fundamental tools in mathematics used to find rates of change. They play a pivotal role in analyzing and understanding functions, particularly when determining the behavior of sinusoidal functions.
By taking the derivative of a function, you can determine its rate of increase or decrease and find critical points where the function might reach local maxima or minima. The derivative of a sine function, such as \(f(x) = \sin(Bx + C)\), is given by \(f'(x) = B \cos(Bx + C)\).
In the context of our function, \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\), the derivative \(D'(t) = 2.8 \cdot \frac{2 \pi}{365} \cos\left(\frac{2 \pi}{365}(t-81)\right)\) helps us locate critical points where the function's slope is zero. These points correspond to the maximum and minimum daylight durations.

Critical Points in Calculus

Critical points in calculus are values in the domain of a function where the derivative is zero or undefined. These points are crucial for finding local maxima, minima, and inflection points.
To locate critical points, you differentiate the function and solve for values of the variable that make the derivative equal to zero. In the context of sinusoidal functions, critical points help us identify significant points in the oscillation, such as peaks and troughs.
For the given function \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\), its derivative, \(D'(t)\), becomes zero when \( \cos\left(\frac{2 \pi}{365}(t-81)\right) = 0\). Solving this yields \(t\) values at 172 and 355, which are the critical points corresponding to the maximum and minimum daylight hours, aligning with the summer and winter solstices, respectively.