(1)There will be 14.21 Hours of daylight on July 19 day 200 according to the graph

(2)Actually there will be 14.17 hours of daylight on day 200. The differenace is 14.21-14.17=.04 hours or 2.4 minutes.

(3) $D'(x)$= 12.1-2.4cos((2pi(x+10))/365)
Plug in 200 since it is the 200th day.
4.8pi/365=sin(420pi)/365=.0026049

(4) The rate at which the number of hours of daylight are changing on July 19 is .0026049 Minutes/day. This means the rate at which the number of hours of daylight changing on July 19 the 200th day- July 19 = D'(200)

(5)D'(0)=4.8pi/365 sin(2pi(x+10))/365=0
=sin(2pi(x+10))/365=npi
=2pi(x+10)/365
x=(365npi-20pi)/2pi
x=(365n-20)/2
n=1 x=172.5 this is June 21st.

(6) You can find the day of the year whent eh number of hours of daylight increase most rapidly by looking at the graph. It is clear to see which days had the most sunlight by looking at the graph- you can see the month with the most daylight and increasing is showin in the light blut, since there is more light blue from early May to the end of August.