Chapter 7.1, Problem 80AE

In Problems 79-84, use the following discussion. The formula $D=24\left[1-\frac{{\cos }^{-1}\left(\tan i\tan \theta \right)}{\pi }\right]$ can be used to approximate the number of hours of daylight D when the declination of the Sun is $i°$ at a location $\theta °$ north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle $\theta$ between the equatorial plane and ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, ${\cos }^{-1}\left(\tan i\tan \theta \right)$ must be expressed in radians.

Approximate the number of hours of daylight in New York, New York ( $40°45\text{'}$ north latitude), for the following dates:

Summer solstice $(i=23.5°)$

Vernal equinox $\left(i=0°\right)$

July 4 $\left(i=22°48\text{'}\right)$