Due: 9/14 MA 251 Group Homework 2 It is possble to define a function of the form

Question

Due: 9/14 MA 251 Group Homework 2 It is possble to define a function of the form L(t) = A + B cos(Ct + D) which gives the length of the day (in hours) where t is the day of the year. 1. Use the fact that the period of L(t) is 365 to find C 2. Use the fact that the longest day of the year is June 22nd (the 173rd day of the year) to find D. 3. Use the fact that on the equinox the day is 12 hours long to find A. 4. Use the fact that on January 9th (the 9th day of the year) there were 9 hours between sunrise and sunset to find B accurate to four decimal places. 5. Use your equation to predict how long the day will be on Easter (Mover B1st is the otlh day of the April5 year) AP

Explanation / Answer

L(t) = A + B cos (Ct + D )

1. period of the function = 2pi/ C

C = 2pi/ 365 = .0172

2. longest day of the year = 173

Ct +D = 173

The cos function is maximum on t=0 without any phase shift.In this case the t=0 has shifted t=D

D = 173 -C*(0) = 173

3. The equinox have 12 hours day .This would be the minimum length of day.It means

the L(t) function which is verticall shifted by A units , passes thorugh this line A .

Bcos(Ct+D) =0 So, L(t) = 12 which gives A = 12

L(t) = 12 +Bcos(0.0172t +173)

4. On 9th jan day is 925 hours

So, L(t) = 9.25 ; t = 9

9.25 = 12 + Bcos( 0.0172*9 + 173)

-2.75 = - 0.99B

B = 2.7778

L(t) = 12 + 2.78cos( 0.0172t + 173)

(5) We have to find the length of day on 95 th day .Plug t= 95

L(95) = 12 -2.7678 = 9.23 Hours

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