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The number of hours and minutes of daylight in Toronto on day 144 is 15 hours and 3 minutes and the number of hours and minutes of daylight in Miami on day 144 is 13 hours and 33 minutes. 5. Explain how the latitude of a location is related to the hours of daylight, and explain how this relationship is illustrated by the differences in the parameters in the two equations. The latitude of a location is related to the hours of daylight, since there are more hours of daylight and a smaller range between the smallest number of hours of daylight and greatest number of hours of daylight as you go south towards the equator.

This relationship is illustrated by the differences in the parameters in the two equations, since a, which is the amplitude of the function, is higher in Toronto (3. 255) than Miami (1. 611). This shows that there is a higher maximum value, as well as a lower minimum value than Miami, which results in a higher amplitude. For the parameter b, the values are the same, since the period is for both functions, which equals to 0. 017. For the parameter c and d, the values are almost the same, since the phase shift, which is very close to the longitude of the two cities, since they move at the same degree from the equator.

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The vertical shift would approximately show the mean number of daylight numbers of the two cities, since a sine function usually starts at the mean value, goes to the maximum value as the x-value increases, and returns to the mean value (and it continues on). As a result, these two values form a coordinate (c, d), which is extremely close to the first point of intersection between the two sinusoidal functions. 6. One factor that affects a region’s growing season is hours of daylight. Toronto’s growing season generally starts when there are 15 or more hours of daylight per day.

If this were the only factor, then what would be the predicted start date and end date of the growing season in Toronto? Explain the method you used to determine these dates. The predicted start date and end date of the growing season in Toronto would be Day 143 and Day 200. The method that is used to determine these dates is through using the TI-84 Plus graphing calculator. The sinusoidal function T(n) = 3. 255 sin[0. 017(n – 78. 351)] + 12. 119 and the line y=15 are put onto the graphing calculator. A graph is drawn from the two equations and the points of intersection are found out using the graphing calculator.

Another way is to find the start date and end date algebraically. 7. Explain how functions T and M, which you determined in part B, questions 1 and 2, could be used to determine the days on which Toronto and Miami have the same number of hours of daylight. Explain the method you would use to determine these dates. The functions T and M could be used to determine the days on which Toronto and Miami have the same number of hours of daylight by putting the two equations on a single grid and find the points of intersection.

The method that I would use to determine these dates would be to insert the equations T(n) = 3. 255 sin[0. 017(n – 78. 351)] + 12. 119 and M(n) = 1. 611 sin[0. 017(n – 78. 843)] + 12. 114 into the TI-84 Plus graphing calculator. Then, a graph would be drawn from the two equations and the points of intersection would be found out using the graphing calculator. As a result, there are two points of intersection that are seen on the grid. The two days on which Toronto and Miami have the same number of hours of daylight are Day 78 and Day 263.

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