In a certain city the number of hours of sunlight on the sum

In a certain city, the number of hours of sunlight on the summer solstice is 15 784 and the number of hours of sunlight on the winter solstice is 8.662. Answer the following questions. Assume that summer solstice occurs on the 172^nd day of the year, and that there are 365 days until the next summer solstice. Find a sinusoidal function of the form y = A sin (omega x - phi) + B that fits the data. A = omega = (Type an exact answer in terms of pi. Use integers or fractions for any numbers in the expression. Simplify your answer.) phi = Type an exact answer in terms of pi. Use integers or fractions for any numbers in the expression. Simplify your answer.) B = Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year. Y approximately equal hours (Round to two decimal places as needed) Draw a graph of the function found in part (a). Choose the correct graph of the function below.

Solution

y = A*sin (wx + phi) + B

A = (number of summer solstice - number of winter solstice)/2

A = (15.784 - 8.662)/2

A = 3.561

T = 2*pi/w

w = 2*pi/365

phi = ?

T = 365 days

dividing in 4 subinterval = 365/4 = 91.25 days

summer solstice occurs on 172nd days

then phi = w*(172 - 91.25) = w*80.75 days

B = vertical shift

B = (number of summer solstice - number of winter solstice)/2

B = (15.784 + 8.662)/2 = 12.223

y = 3.561*sin ((2*pi/365)(x - 80.75)) + 12.223

B.

on 91st day of the year

y = 3.561*sin ((2*pi/365)(91 - 80.75)) + 12.223

y = 12.848 hrs