Answer
There will be about 10 hours of daylight on November 5th and February 3rd.
Work Step by Step
$h = \frac{35}{3}+\frac{7}{3}sin~\frac{2\pi~x}{365}$
We can find $x$ when $h=10$:
$h = \frac{35}{3}+\frac{7}{3}sin~\frac{2\pi~x}{365}$
$3h = 35+7sin~\frac{2\pi~x}{365}$
$sin~\frac{2\pi~x}{365} = \frac{3h-35}{7}$
$sin~\frac{2\pi~x}{365} = \frac{(3)(10)-35}{7}$
$sin~\frac{2\pi~x}{365} = -\frac{5}{7}$
Note that $arcsin(-\frac{5}{7}) = -0.7956$
There will be two values for x, where $0 \leq x \lt 365$, such that $sin\frac{2\pi~x}{365} = -\frac{5}{7}$
These two values occur when $\frac{2\pi~x}{365} = \pi + 0.7956$ and $\frac{2\pi~x}{365} = 2\pi - 0.7956$. We can find the value of $x$ in both cases.
We can find $x$ when $\frac{2\pi~x}{365} = \pi+0.7956$:
$\frac{2\pi~x}{365} = \pi+0.7956$
$x = \frac{(\pi+0.7956)(365)}{2\pi}$
$x = 229~days$
We can find the date that is 229 days after March 21st:
March 31st: 10 days
April 30th: 30 days + 10 days = 40 days
May 31st: 31 days + 40 days = 71 days
June 30th: 30 days + 71 days = 101 days
July 31st: 31 days + 101 days = 132 days
August 31st: 31 days + 132 days = 163 days
September 30th: 30 days + 163 days = 193 days
October 31st: 31 days + 193 days = 224 days
November 5th: 5 days + 224 days = 229 days
We can find $x$ when $\frac{2\pi~x}{365} = 2\pi-0.7956$:
$\frac{2\pi~x}{365} = 2\pi-0.7956$
$x = \frac{(2\pi-0.7956)(365)}{2\pi}$
$x = 319~days$
We can find the date that is 319 days after March 21st:
March 31st: 10 days
April 30th: 30 days + 10 days = 40 days
May 31st: 31 days + 40 days = 71 days
June 30th: 30 days + 71 days = 101 days
July 31st: 31 days + 101 days = 132 days
August 31st: 31 days + 132 days = 163 days
September 30th: 30 days + 163 days = 193 days
October 31st: 31 days + 193 days = 224 days
November 30th: 30 days + 224 days = 254 days
December 31st: 31 days + 254 days = 285 days
January 31st: 31 days + 285 days = 316 days
February 3rd: 3 days + 316 days = 319 days
There will be about 10 hours of daylight on November 5th and February 3rd.
