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. 14 hours (round to the nearest hour as needed.) d. how many hours of daylight are there on the shortest day of the year? 10 hours (round to the nearest hour as needed.) e. graph the function for one period, starting on january 1. use the graphing tool to graph the function. click to enlarge graph (for any answer boxes shown with the grapher, type an exact answer. type the word pi to insert the symbol π 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. 14 hours (round to the nearest hour as needed.) d. how many hours of daylight are there on the shortest day of the year? 10 hours (round to the nearest hour as needed.) e. graph the function for one period, starting on january 1. use the graphing tool to graph the function. click to enlarge graph (for any answer boxes shown with the grapher, type an exact answer. type the word pi to insert the symbol π 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 $y = 2\sin\left(\frac{2\pi}{365}(x - 81)\right)+12$, where $A = 2$, $B=\frac{2\pi}{365}$, $C = 81$, and $D = 12$. The range of the sine function $\sin(u)$ is $[- 1,1]$.

Step2: Find the minimum value of the given function

To find the number of hours of daylight on the shortest day, we need to find the minimum value of the function $y = 2\sin\left(\frac{2\pi}{365}(x - 81)\right)+12$. Since the minimum value of $\sin\left(\frac{2\pi}{365}(x - 81)\right)$ is $-1$. Substitute $\sin\left(\frac{2\pi}{365}(x - 81)\right)=-1$ into the function: [ \begin{align*} y&=2\times(-1)+12\ y&=-2 + 12\ y&=10 \end{align*} ]

Answer:

10 hours