Question

The number of hours of daylight measured in one year in Ellenville can be modeled by a sinusoidal function. During 2006, (not a leap year), the longest day occurred on June 21 with 15.7 hours of daylight. The shortest day of the year occurred on December 21 with 8.3 hours of daylight. Write a sinusoidal equation to model the hours of daylight in Ellenville.

Answer 1

Answer:

$f ( t ) = 3.7*sin ( 0.01736*t ) + 12$

Step-by-step explanation:

Solution:-

- We are to model a sinusoidal function for the number of hours of daylight measured in one year in Ellenville.

- We will express a general form of a sinusoidal function [ f ( t ) ] as follows:

                                $f ( t ) = A*sin ( w*t ) + c$

Where,

                  A: The amplitude of the hours of daylight

                  w: The angular frequency of occurring event

                  c: The mean hours of daylight

                  t: The time taken from reference ( days )

- We are given that the longest day [ $f ( t_m_a_x )$ ] occurred on June 21st and the shortest day [ $f ( t_m_i_n )$ ] on December 21st.

- The mean hours of daylight ( c ) is the average of the maximum and minimum hours of daylight as follows:

                              $c = \frac{f(t_m_a_x ) +f(t_m_i_n ) }{2} \\\\c = \frac{15.7 + 8.3}{2} = \frac{24}{2} \\\\c = 12$

- The amplitude ( A ) of the sinusoidal function is given by the difference of either maximum or minimum value of the function from the mean value ( c ):

                              $A = f ( t_m_a_x ) - c\\\\A = 15.7 - 12\\\\A = 3.7$

- The frequency of occurrence ( w ) is defined by the periodicity of the function. In other words how frequently does two maximum hours of daylight occur or how frequently does two minimum hours of daylight occur.

- The time period ( T ) is the time taken between two successive maximum duration of daylight hours. We were given the longest day occurred on June 21st and the shortest day occurred on December 21st. The number of days between the longest and shortest day will correspond to half of the time period ( 0.5*T ):

                              $0.5*T = 7 + 31 + 31 + 30 +31 +30 +21\\\\T = 2* [ 181 ] \\\\T = 362 days$

- The angular frequency ( w ) is then defined as:

                              $w = \frac{2\pi }{T} = \frac{2\pi }{362} \\\\w = 0.01736$

- We will now express the model for the duration of daylight each day as function of each day:

                              $f ( t ) = 3.7*sin ( 0.01736*t ) + 12$