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

question 325

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) \(y = 11.23 \sin \left( \pi x - \frac { 2 \pi } { 3 } \right) + 8.03\)

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Definitions:

Pearson Correlation

A statistical measure that calculates the strength and direction of the linear relationship between two quantitative variables.

Pearson Correlation

A measure of linear correlation between two variables, ranging from -1 to 1, where 1 means a perfect positive linear relationship, -1 means a perfect negative linear relationship, and 0 indicates no linear relationship.

Non-Directional

Refers to a hypothesis that predicts a change or difference but does not specify the direction of the change.

Sample Size

The number of observations or data points collected from a subset of a larger population for the purpose of statistical analysis.