85. The number of hours of daylight in Boston is given by \[ y=3 \sin \frac{2 \pi}{365}(x-79)+12 \text {, } \] where \( x \) is the number of days after January 1 . a. What is the amplitude of this function? b. What is the period of this function? c. How many hours of daylight are there on the longest day of the year? d. How many hours of daylight are there on the shortest day of the year? e. Graph the function for one period, starting on January 1 .

The amplitude of the function \( y = 3 \sin \frac{2 \pi}{365}(x-79) + 12 \) is 3. This means that the number of hours of daylight varies by 3 hours above and below the average daylight of 12 hours (which is the vertical shift of the function). So the range of hours of daylight is from 9 hours to 15 hours. The period of the function is 365 days, derived from the coefficient \( \frac{2\pi}{365} \). This indicates that the cycle of daylight repeats annually, giving you a full year's experience from one January 1 to the next. To find out the hours of daylight on the longest day of the year, set \( \sin \) to 1, resulting in \( y = 3(1) + 12 = 15 \) hours on the summer solstice around June 21. Conversely, on the shortest day when \( \sin \) equals -1, \( y = 3(-1) + 12 = 9 \) hours, which occurs around December 21. To graph the function for one period, plot the sine wave starting at \( x = 0 \) (January 1) rising to 15 hours at \( x = 172 \) (around June 21), dipping to 9 hours at \( x = 355 \) (around December 21), and returning back to 12 hours on January 1 of the next year. The wave will have peaks and valleys reflecting the cyclical nature of seasons and daylight changes throughout the year.