the number of hours of daylight on a given day in city a is given by the following function, where x is the…

the number of hours of daylight on a given day in city a is given by the following function, where x is the number of days after january 1. y = 2 sin((2π/365)(x - 81)) + 12 use this function to answer parts a through e. a. what is the amplitude of this function? 2 (simplify your answer.) b. what is the period of this function? 365 (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. how many hours of daylight are there on the longest day of the year? 14 hours (round to the nearest hour as needed.) d. how many hours of daylight are there on the shortest day of the year? hours (round to the nearest hour as needed.)

the number of hours of daylight on a given day in city a is given by the following function, where x is the number of days after january 1. y = 2 sin((2π/365)(x - 81)) + 12 use this function to answer parts a through e. a. what is the amplitude of this function? 2 (simplify your answer.) b. what is the period of this function? 365 (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. how many hours of daylight are there on the longest day of the year? 14 hours (round to the nearest hour as needed.) d. how many hours of daylight are there on the shortest day of the year? hours (round to the nearest hour as needed.)

Answer

Explanation:

Step1: Recall the general form of a sinusoidal function

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$. The amplitude is $|A|$, the period is $\frac{2\pi}{|B|}$, the maximum value is $D + |A|$ and the minimum value is $D-|A|$. For the function $y = 2\sin(\frac{2\pi}{365}(x - 81))+12$, we have $A = 2$, $B=\frac{2\pi}{365}$, $C = 81$ and $D=12$.

Step2: Find the number of hours of daylight on the shortest day

The minimum value of the sine - function is $- 1$. Substitute $\sin(\frac{2\pi}{365}(x - 81))=-1$ into the function $y = 2\sin(\frac{2\pi}{365}(x - 81))+12$. $y=2\times(-1)+12$. $y=-2 + 12$. $y = 10$.

Answer:

10