11th gradeCalculas
posted by Anonymous .
The number of hours of daylight D depends upon the latitude and the day t of the year and is given by the equation :
D(t)=12+Asin(((2pi)/(365))(t80))
where A depends only on the latitude (and not t). for latitude 30 degress, A is about 2.3
1. When is the number of hours of daylight the greatest?
2.When is the number of hours of daylight the least?
3. When is the number of hours of daylight increasing at a rate of 2 minutes per day?
4. when is the number of hours of daylight decreasing at a rate of 2 min per day?
Can any one help me with any of these?
thanks a lot!

Take the derivitive of D with respect to t. Set to zero, solve for the solutions. Those will be max/min day lengths. You can use the second derivitative to find out which.
for 3,4, set dD/dt equal to 2/60 and solve for t. 
wait to take the derivative, should i multiple out the stuff inside the ()?

dD/dt = A(2pi/365)*cos [(2pi/365)*(t80)]
=0 for maximum or minimum
(2pi/365)*(t80) = n*pi/2
when n=1 gives maximum , then (t80) = 365/4, for minimum n = 3
and t80 = 3*365/4
(3) = dD/dt = +2/60
(4) = dD/dt = 2/60