Answer
The least number of hours of daylight will occur on December 20th.
Work Step by Step
$h = \frac{35}{3}+\frac{7}{3}sin~\frac{2\pi~x}{365}$
The least number of hours of daylight will occur when $sin~\frac{2\pi~x}{365} = -1$. We can find $x$ when $sin~\frac{2\pi~x}{365} = -1$:
$sin~\frac{2\pi~x}{365} = -1$
$\frac{2\pi~x}{365} = arcsin(-1)$
$\frac{2\pi~x}{365} = \frac{3\pi}{2}$
$x = \frac{(3)(365)}{4}$
$x = 274~days$
We can find the date that is 274 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 20th: 20 days + 254 days = 274 days
The least number of hours of daylight will occur on December 20th.
