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 - 80))+12. use this function to answer parts a through e. 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 - 80))+12. use this function to answer parts a through e. 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: Analyze the sine - function properties

The general form of a sine function is $y = A\sin(B(x - C))+D$. In the given function $y = 2\sin\left(\frac{2\pi}{365}(x - 80)\right)+12$, we have $A = 2$, $B=\frac{2\pi}{365}$, $C = 80$, and $D = 12$.

Step2: Recall the range of the sine function

The range of the sine function $\sin(u)$ is $[- 1,1]$.

Step3: Find the minimum value of the given function

To find the number of hours of daylight on the shortest day, we consider when $\sin\left(\frac{2\pi}{365}(x - 80)\right)=-1$. Substitute $\sin\left(\frac{2\pi}{365}(x - 80)\right)=-1$ into the function $y = 2\sin\left(\frac{2\pi}{365}(x - 80)\right)+12$. $y=2\times(-1)+12$.

Step4: Calculate the result

$y=-2 + 12=10$.

Answer:

10 hours