Calculus Modeling Formulas-Seasons and Months

Oct 2015
1
0
NY
The following formula is a mathematical model for the number of hours of daylight on the t day of the year: L(t)=12+2.9sin[2pi/3659(t-80)]

1. Use this model to compare how the number of hours of daylight is increasing from a) March 21st-June 21st b) June 21st-Sept 21st c) Sept 21st-Dec 21st d) Dec 21st-March 21st

2. Use this model to compare how the number of daylight is increasing from a) Jan 6th-Feb 6th b) Feb 6th-Mar 6th and so on....

During which time period is the amount of light increasing the most? What is the significance of "12", "2.9 (0 being the equator and 12 being the North Pole), "2pi", "2pi/365", "80", and "t-80" in the model formula?

THANKS SO MUCH!!! EVEN AN EXPLANATION ON HOW TO DO IT AND DOING ONLY THE FIRST FOR EACH WILL BE A REALLY BIG HELP.
 

jks

Jul 2012
642
99
DFW Area
Hi Nzerb, and welcome to the forum.

I assume that you meant:

\(\displaystyle L(t)=12+2.9 \sin \left(\frac{2\pi}{365} \cdot (t-80) \right)\)

Use this model to compare how the number of hours of daylight is increasing from a) March 21st-June 21st b) June 21st-Sept 21st ...
I am having a hard time understanding exactly what questions 1. and 2. are asking for. Since the word 'increasing' is used, I think you need to take the derivative of $L(t)$ and evaluate the result at the dates given and compare. However, since intervals are given, evaluating the integral of $L(t)$ may be involved but I doubt it.

During which time period is the amount of light increasing the most?
You can add the derivative values for the interval start and end points in 2. (I assume), to find the most positive result. As a check, it is pretty easy to calculate the day, t, of the maximum increase. To do this, take the derivative, $L'(t)$ and find the maximum (it should be ~2.995 minutes/day). If you need help with taking the derivative, please reply and I or another member can help.

What is the significance of "12", "2.9 (0 being the equator and 12 being the North Pole), "2pi", "2pi/365", "80", and "t-80" in the model formula?
First, let's start with the coefficient of the sine term, 2.9. It is for approximately 40 degrees latitude (north, as we will see from the t-80 term). It is latitude dependent and I agree with 0 for the equator, but it seems to me that 12 is for the Arctic Circle, not the North Pole since at the latitude of the Arctic Circle there will be 0 hours of sunlight on the day of the winter solstice and 24 hours of sunlight on the day of the summer solstice.

At higher latitudes the number will be higher, tending to a very large number for the North Pole, with numbers greater than 24 truncated to 24 and numbers less than 0 truncated to 0.

With the coefficient of the sine term at 0 for the equator, can you see the significance of the 12?

Since the sine function repeats every $2\pi$ units and a year is roughly 365 days long, can you see the significance of the $2\pi$ and $\large \frac{2\pi}{365}$ terms?

To see the significance of the 80 and the t-80, consider when the number of hours of sunlight is a minimum, on the day of the winter solstice. When is the sine function a minimum? When is the winter solstice for the northern hemisphere?

Does this help?


Note: The t values for the dates are given here.