Question

The number of hours of daylight at any point on Earth fluctuates throughout the year. In the northern hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At \(40^{\circ}\) north latitude, the length of a day is approximated by $$D(t)=12-3 \cos \left(\frac{2 \pi(t+10)}{365}\right),$$ where \(D\) is measured in hours and \(0 \leq t \leq 365\) is measured in days, with \(t=0\) corresponding to January 1 a. Approximately how much daylight is there on March 1 \((t=59) ?\) b. Find the rate at which the daylight function changes. c. Find the rate at which the daylight function changes on March 1. Convert your answer to units of min/day and explain what this result means. d. Graph the function \(y=D^{\prime}(t)\) using a graphing utility. e. At what times of year is the length of day changing most rapidly? Least rapidly?

Short answer

Answer: The length of a day changes most rapidly around the Spring Equinox (around March 21st) and Fall Equinox (around September 22nd). It changes least rapidly around the Summer Solstice (around June 21st) and the Winter Solstice (around December 21st).

Step-by-step solution

a. Daylight hours on March 1 (t=59)#

To find the number of daylight hours on March 1st, we need to plug in t=59 into the given function: $$D(59)=12-3 \cos \left(\frac{2 \pi(59+10)}{365}\right)$$ Compute this value to get the number of daylight hours: $$D(59) \approx 11.61$$ So, there are approximately 11.61 hours of daylight on March 1st.

b. Rate of change of the daylight function#

To find the rate of change of the daylight function, we need to take its derivative with respect to t: $$D^{\prime}(t) = \frac{d}{dt}\left[12-3 \cos \left(\frac{2 \pi(t+10)}{365}\right)\right]$$ Apply the chain rule and compute the derivative: $$D^{\prime}(t) = 3\left(\frac{2\pi}{365} \right) \sin\left(\frac{2\pi(t+10)}{365}\right)$$ This is the rate at which the daylight function changes with respect to time.

c. Rate of change of the daylight function on March 1 and conversion to min/day#

Now we need to find the rate of change of the daylight function on March 1 by substituting t=59 in the derivative function: $$D^{\prime}(59) = 3\left(\frac{2\pi}{365}\right) \sin\left(\frac{2\pi(59+10)}{365}\right)$$ Compute the value: $$D^{\prime}(59) \approx 0.0732$$ To convert this to min/day, we multiply the result by 60 (minutes in an hour): $$ 0.0732 \, \text{hr/day} \times \frac{60 \, \text{min}}{1\, \text{hr}} \approx 4.39 \, \text{min/day}$$ This means that on March 1, the length of a day is increasing at a rate of approximately 4.39 minutes per day.

d. Graph of the derivative function y = D'(t)#

This task needs to be done with a graphing calculator or any software that plots graphs. The function to be plotted is: $$y = D^{\prime}(t) = 3\left(\frac{2\pi}{365} \right) \sin\left(\frac{2\pi(t+10)}{365}\right)$$

e. Times when the length of day changes most and least rapidly#

To determine when the length of day has the maximum and minimum rate of change, we should analyze the graph of the derivative function. Local maxima and minima occur when the derivative function is equal to 0. - When \(D^{\prime}(t) = 0\) and changing from positive to negative, we have local maxima (the length of day is increasing most rapidly). - When \(D^{\prime}(t) = 0\) and changing from negative to positive, we have local minima (the length of day is decreasing most rapidly). From the graph, we can observe the maxima and minima points, and approximate their location to the specific time of the year. The approximate time when the length of day changes most rapidly is near: - Spring Equinox (around March 21st) - Fall Equinox (around September 22nd) The length of day changes least rapidly around: - Summer Solstice (around June 21st) - Winter Solstice (around December 21st)

Key concepts

Derivative

The concept of derivative is all about finding the rate at which something changes. It's like asking how fast a car is going at a particular moment instead of its average speed over a journey. The mathematical process of differentiation helps us find this instantaneous rate of change.

For the daylight function, the derivative, denoted as \(D'(t)\), tells us exactly how the length of the day is changing each day. In trigonometric terms, we use sinusoidal functions, like sine and cosine, because they naturally model periodic phenomena like daylight through the year. Understanding derivatives is key to interpreting how dynamic changes unfold over time.

In this exercise, the derivative of the daylight function with respect to time \(t\) is derived using the chain rule, which is a fundamental rule in calculus used to differentiate compositions of functions.

Rate of Change

The rate of change explains how one quantity changes in relation to another. It's like asking, "How fast is the length of the day changing?" When dealing with continuous functions, as in this exercise, the rate of change gives us a precise measure of how a quantity varies at any specific point in time.

In the context of daylight, examining the rate at which daylight changes involves differentiating the daylight function. By computing the derivative \(D'(t)\), we can see how the daylight lengthens or shortens as days go by. On March 1, for instance, the rate of change is approximately 4.39 minutes per day.

To find this, we compute the derivative at \(t=59\) and convert the rate from hours per day to minutes per day. This allows us to understand how rapidly daylight increases as we approach the equinoxes in March and September.

Daylight

Daylight is the period of time each day when the sun is above the horizon, offering natural light. The duration of daylight varies widely due to the axial tilt of the Earth and its orbit around the Sun. This cyclic pattern means that some parts of the year have longer days while others have shorter days. Latitude significantly influences how much daylight a region receives, which is why this exercise uses a model to approximate daylight hours at a 40° north latitude.

This model, given by \(D(t) = 12 - 3 \cos\left(\frac{2\pi(t+10)}{365}\right)\), uses a cosine function due to its ability to model periodic changes smoothly. The formula captures the gradual increase and decrease in daylight hours seen throughout the year, calculating changes based on days elapsed since the start of the year. Understanding such a model helps interpret how and why daylight shifts, especially in relation to equinoxes and solstices, where these changes are most noticeable.

Seasonal Variation

Seasonal variation refers to the predictable and recurrent changes seen in Earth’s environment, driven by its tilt and orbit around the sun. These changes result in the four seasons and govern phenomena like temperature fluctuations, plant life cycles, and importantly, daylight variation.

The length of day changes as these seasons shift, with the most dramatic changes occurring at the equinoxes, when day and night are almost equal, and the least change at the solstices, when day or night reaches its extreme. The cosine function within the given daylight model helps illustrate these seasonal variations by providing a mathematical way to simulate the transition from long summer days to short winter days, and vice versa.

Understanding seasonal variation allows us to appreciate the natural cycles of Earth, which are essential for ecological balance and have a profound impact on human activities and natural ecosystems.