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Math Help - Minutes of Daylight Per Day

  1. #1
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    Minutes of Daylight Per Day

    The number of minutes per day L(d) at 40 degrees North latitude is modeled by the function
    L(d)=167.5 sin[(2pi/366)(d-80)] + 731
    where d is the number of days after the beginning of 1996. (For Jan 1,1996 d=1; and for Dec 31 19996 d=366 since 1996 was a leap year)

    A) Which day has the most minutes of daylight? Justify your answer
    B) What is the average number of minutes of daylight in 1996? Justify your answer
    C) What is the total number of minutes of daylight in 1996? Justify your answer

    So for A i think you need to take dL/dt and find the max but i don't know how to take this derivative or how to set it equal to 0.

    For B would you subtract the number of minutes of the first day from the minutes from the last day and divide them by (366-1)?

    For C would you do a summation?
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  2. #2
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    Quote Originally Posted by rawkstar View Post
    The number of minutes per day L(d) at 40 degrees North latitude is modeled by the function
    L(d)=167.5 sin[(2pi/366)(d-80)] + 731
    where d is the number of days after the beginning of 1996. (For Jan 1,1996 d=1; and for Dec 31 19996 d=366 since 1996 was a leap year)

    A) Which day has the most minutes of daylight? Justify your answer
    B) What is the average number of minutes of daylight in 1996? Justify your answer
    C) What is the total number of minutes of daylight in 1996? Justify your answer

    So for A i think you need to take dL/dt and find the max but i don't know how to take this derivative or how to set it equal to 0.

    the maximum value of the sine function ...

    \textcolor{red}{\sin\left(\frac{\pi}{2}\right) = 1}

    so ... wouldn't the maximum occur when

    \textcolor{red}{\frac{2\pi}{366}(d-80) = \frac{\pi}{2}} ?

    For B would you subtract the number of minutes of the first day from the minutes from the last day and divide them by (366-1)?

    the average number of daylight minutes would be 731 ... why is that?

    For C would you do a summation?

    \textcolor{red}{\int_0^{366} L(d) \, dd}
    ...
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  3. #3
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    ok i got 171.5 for A
    and i have no idea how u did B
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  4. #4
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    I'm surprised that you would be taking Calculus, asked to do a problem like this and not know the derivative of sin(x). It is cos(x).

    There are enough days in a year that you can accurately approximate this by a continuous function with d going from 0 to 366.
    Also you should know that the "average" of a continuous function, between x= a and x= b is \frac{\int_a^b f(x) dx}{b- a}. For that you will need to know that the anti-derivative (integral) of sin(x) is -cos(x).

    To find the total hours of sunlight, once you have found the average, multiply by b- a= 366- 0. Of course, that is the same as the integral, \int_a^b f(x)dx.
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    i know the derivative of sin is cos, im not an idiot
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  6. #6
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    ok, i thought about this and figured A & B out
    what i still don't understand is why C is from 0 to 366instead of it being from 1 to 366 since it starts at d=1
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  7. #7
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    Quote Originally Posted by rawkstar View Post
    ok, i thought about this and figured A & B out
    what i still don't understand is why C is from 0 to 366instead of it being from 1 to 366 since it starts at d=1
    ... according to that logic, a day would start at 1:00 am and end 23 hrs later?
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