Solve the Problem $11.23$ Hours of Sunlight at the Summer Solstice And

Multiple Choice

Solve the problem.
-The number of hours of sunlight in a day can be modeled by a sinusoidal function. In the northern hemisphere, the longest day of the year occurs at the summer solstice and the shortest day occurs at the winter solstice. In 2000, these
Dates were June 22 (the 172nd day of the year) and December 21 (the 356th day of the year) , respectively. A town experiences $11.23$ hours of sunlight at the summer solstice and $8.03$ hours of sunlight at the winter solstice. Find a sinusoidal function $y = A \sin ( \omega x - \varphi ) + B$ that fits the data, where $x$ is the day of the year. (Note: There are 366 days in the year $2000 . )$

A) $y = 1.6 \sin \left( \frac { \pi } { 183 } x - \frac { 161 \pi } { 366 } \right) + 9.63$
B) $y = 11.23 \sin \left( \frac { 172 \pi } { 356 } x - \frac { 2 \pi } { 3 } \right) + 9.63$
C) $y = 1.6 \sin \left( \frac { \pi } { 183 } x - \frac { 2 \pi } { 3 } \right) + 9.63$
D) $y = 11.23 \sin \left( \pi x - \frac { 2 \pi } { 3 } \right) + 8.03$