A city averages 14 hours of daylight in June, 10 hours in December, and 12 hours in March and September. We know that the number of hours of daylight varies sinusoidally over one year.
1). Write an expression for n, the number of daylight hours as a cosine function of t. Let t be in months and t = 0 correspond to January.
2) What is the average amount of daylight hours in August

The expression for the number of hours of daylight as a cosine function is n = 2cos((2π/12)(t-6)) + 12. The average amount of daylight hours in August is approximately 13.73 hours.

Step-by-step explanation:

To write an expression for the number of hours of daylight (n) as a cosine function of time (t), we need to determine the amplitude, period, and phase shift of the function. The amplitude of the function is half of the difference between the maximum and minimum values, which is 2 hours (14 - 10 = 4, and 4 / 2 = 2). The period of the function is 12 months, as the pattern repeats yearly. The phase shift is calculated based on the starting month. Since January (t = 0) corresponds to the minimum daylight in December, the phase shift is 6 months (12 / 2 = 6).

Therefore, the expression for the number of hours of daylight as a cosine function would be:

n = 2cos((2π/12)(t-6)) + 12

To find the average amount of daylight hours in August (t = 7), we substitute t = 7 into the equation:

n = 2cos((2π/12)(7-6)) + 12

n = 2cos((2π/12)(1)) + 12

n = 2cos(π/6) + 12

n = 2(√3/2) + 12

n = √3 + 12

n ≈ 13.73 hours