\begin{tabular}{l} 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 5 . \ $$ y=3 $$ sin $$ \left(\frac{2 \pi}{365}(x-79) ight)+12 $$ \ Use this function to answer parts a through e. \ a. What is the amplitude of this function? \ 3 (Simplify your answer.) \ b. Whathis the period of this function? \ 365 \ (Simplify your answer. Type an exact answer, using $$ \pi $$ 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? \ $$ \square $$ hours \ (Round to the nearest hour as needed.) \ \hline\end{tabular}

Question

\begin{tabular}{l} 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 5 . \ $$ y=3 $$ sin $$ \left(\frac{2 \pi}{365}(x-79)ight)+12 $$ \ Use this function to answer parts a through e. \ a. What is the amplitude of this function? \ 3 (Simplify your answer.) \ b. Whathis the period of this function? \ 365 \ (Simplify your answer. Type an exact answer, using $$ \pi $$ 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? \ $$ \square $$ hours \ (Round to the nearest hour as needed.) \ \hline\end{tabular}

UpStudy AI · Accepted Answer

We are given the function 
  y = 3 sin((2π/365)(x – 79)) + 12.

a. The amplitude is the absolute value of the coefficient of the sine function, which is |3| = 3.

b. The period of a sine function written as sin(Bx) is (2π)/B. Here, B = 2π/365, so the period is
  (2π)/(2π/365) = 365.

c. The number of hours of daylight on the longest day is the maximum value of the function. Since the sine function ranges between –1 and 1, its maximum is 1. Thus,
  Maximum y = 3(1) + 12 = 15.
Rounded to the nearest hour, there are 15 hours of daylight on the longest day.
a. The amplitude is 3. b. The period is 365 days. c. There are 15 hours of daylight on the longest day.

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