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Exercise 1.3.27
The city of New Orleans is located at latitude 30 N. Use Figure 9 to find a function that models the number of hours of daylight in New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 am and sets at 6:18 pm in New Orleans.
Answers
Notice that each curve resembles a shifted and stretched sine function. By looking at the red curve we see that, at the latitude of New Orleans, daylight lasts about 14 hours on June 21 and 10 hours on December 21, so the amplitude of the curve is
Because there are about 365 days in a year, the period of our model should be 365. But the period of is , so the horizontal stretching factor is We also notice that the curve begins its cycle on March 21, the 80th day of the year, so we have to shift the curve 80 units to the right. In addition, we shift it 12 units upward. Therefore we model the length of daylight in New Orleans on the th day of the year by the function:
