Question

Daylight function for \(40^{\circ}\) N 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 the number of hours between sunrise and sunset. 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. \(D(81)=12\) and \(D(264) \approx 12\) (corresponding to the equinoxes).

Short answer

Question: Verify the properties of the daylight function \(D(t) = 2.8\sin\left(\frac{2 \pi}{365}(t-81)\right)+12\), which represents the number of hours between sunrise and sunset at \(40^{\circ}N\) latitude. Show that: a. The period of the function is 365 days. b. The maximum and minimum values of the function are 14.8 and 9.2, occurring approximately at t=172 and t=355, respectively. c. The function takes a value of 12 at t=81 and t=264.

Step-by-step solution

Find the period of the function

To find the period of the function, we need to analyze the sine function inside. The general form of a sine function is \(A\sin(Bx+C)+D\). The period of \(\sin(Bx+C)\) is \(\frac{2 \pi}{B}\). In this case, the sine function is given by \(2.8\sin\left(\frac{2 \pi}{365}(t-81)\right)\). Therefore, B is \(\frac{2 \pi}{365}\). Plugging this value into the formula, we find the period to be \(365\) days, as required.

Find the maximum and minimum values

To find the maximum and minimum values of the function, we need to look for the values of \(D(t)\) when the sine function inside reaches its maximum and minimum values. Since the sine function varies between -1 and 1, the maximum value of \(D(t)\) occurs when the sine function is equal to 1, and the minimum occurs when it is equal to -1. For the maximum value, we have: \(D(t_{max}) = 2.8\cdot1 + 12 = 14.8\) For the minimum value, we have: \(D(t_{min}) = 2.8\cdot(-1) + 12 = 9.2\) Now, to find the approximate values of \(t\) when these maxima and minima occur, we need to solve the following equations: $\frac{2\pi}{365}(t_{max} - 81) = \pi / 2 \\ \frac{2\pi}{365}(t_{min} - 81) = \pi / -2 $ Solving these equations, we get \(t_{max} \approx 172\) and \(t_{min} \approx 355\). This confirms part (b) of the problem.

Evaluate the function at t=81 and t=264

Now, we need to verify if \(D(81) = 12\) and \(D(264) \approx 12\). To do this, we will simply plug these values of \(t\) into the given function: \(D(81) = 2.8\sin\left(\frac{2\pi}{365}(81-81)\right)+12 = 2.8\sin(0) + 12 = 12\) \(D(264) = 2.8\sin\left(\frac{2\pi}{365}(264-81)\right) + 12 \approx 2.8\cdot\sin(2\pi\cdot\frac{183}{365}) + 12 \approx 12\) Thus, our calculations verify the properties stated in part (c) of the problem.

Key concepts

Sinusoidal Function

A sinusoidal function is a mathematical function that describes a smooth, periodic oscillation. It can be represented by the sine or cosine function and is widely used in physics, engineering, and natural phenomena modeling. The function expressed here is a sine wave, a common type of sinusoidal function.

In the exercise, the function given is:

• \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\)

This function models the hours of daylight over a year for a location at 40° latitude.

The term inside the sine function, \(\frac{2 \pi}{365}(t-81)\), represents the angle in radians and controls the frequency of the oscillation, dictated by the annual transition of daylight. The amplitude, or height, of this wave is 2.8, showing the extent of variation in daylight hours. Understanding sinusoidal functions is essential for predicting and explaining biological, environmental, and physical systems behavior.

Periodicity

The concept of periodicity refers to the repeating cycle of a function over a specific interval. For sinusoidal functions, this cycle is known as the period.

In our daylight function, the period is particularly important because it determines how often the cycle of daylight variation repeats. The given function:

• \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\)

shows a period of 365 days, representing the Earth's orbit around the Sun. For a sine function, the period \(T\) is given by \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(t\) inside the sine function.

Calculating the period, we use the formula:

• \(T = \frac{2\pi}{\left(\frac{2\pi}{365}\right)} = 365\)

This periodicity captures the cyclical nature of seasons dictating daylight duration, aligning naturally with our experiences of longer days in summer and shorter days in winter.

Maximum and Minimum Values

Maximum and minimum values in a function refer to the highest and lowest points the function can reach. These values are essential in understanding the range of behaviors the function can exhibit.

For our daylight function:

• \(D(t) = 2.8 \sin\left(\frac{2 \pi}{365}(t-81)\right) + 12\)

we find the maximum and minimum by examining when the sine function reaches its extremities, namely \(1\) for the maximum and \(-1\) for the minimum.

• Maximum Daylight: \(D(t) = 2.8 \cdot 1 + 12 = 14.8\)
• Minimum Daylight: \(D(t) = 2.8 \cdot (-1) + 12 = 9.2\)

These values correspond to the longest and shortest periods of daylight, occurring during solstices. Solving for specific \(t\), we find these approximately at \(t \approx 172\) and \(t \approx 355\), marking the summer and winter solstices respectively.

Equinoxes and Solstices

Equinoxes and solstices are significant astronomical events that affect daylight hours.

• Equinoxes: These occur twice a year when day and night are almost equal lengths. In terms of our function, they happen when \(t = 81\) and \(t = 264\), corresponding to 12 hours of daylight, showing a balanced transition between seasons.
• Solstices: Solstices mark the extremes of daylight variation. The summer solstice, around \(t \approx 172\), is when daylight is the longest. Conversely, the winter solstice, around \(t \approx 355\), features the shortest daylight hours.

Understanding these points in the cycle provides insight into the natural rhythm of daylight changes. They serve as markers for seasonal change and are deeply intertwined with cultural and agricultural practices worldwide.