Originally Posted by

**rawkstar** 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 ...

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

so ... wouldn't the maximum occur when

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

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?

$\displaystyle \textcolor{red}{\int_0^{366} L(d) \, dd}$