Question

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

(a) Summer solstice$\left(i=23.5°\right)$

(b) Vernal equinox$\left(i=0°\right)$

(c) July$4\left(i=22°{48}^{\text{'}}\right)$

Short answer

(a) The number of hours of daylight in summer solstice is,$14$hours and $53$minutes.

(b) The number of hours of daylight in the vernal equinox is $12$hours.

(c) The number of hours of daylight in July $4$is $14$hours and $49$minutes.

Step-by-step solution

Step 1. Given data

We have to find the number of hours of daylight in New York , New York$40°45\text{'}$north latitude in Summer solstice $i=23.5°$,Vernal equinox $(i=0°)$and on July $4$$(i=22°48\text{'})$

Here, we have to convert the value of north latitude into degree by dividing the minute part by $60$

$\theta =45.75°$

Step 2. Number of hours of daylight in the summer solstice 

(a) To find the number of hours of daylight we have to use the formula,

$D=24\left[1-\frac{{\cos }^{-1}(\tan i\cdot \tan \theta )}{\pi }\right].....\left(1\right)$

Substitute $i=23.5^{°}$and localid="1647587765462" $\theta =40.75°$in this equation,

localid="1647590818723" $\begin{array}{rl}D& =24\left[1-\frac{{\cos }^{-1}(\tan (23.5)\tan (40.75\left)\right)}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}(0.43\times 0.86)}{\pi }\right]\\ & =24[1-\frac{{\cos }^{-1}\left(0.3698\right)}{\pi }]\\ & =24\left[1-\frac{\left(0.3796\right)\pi }{\pi }\right]\\ =& 24[1-0.3796]\\ =& 14.89\end{array}$

Therefore, the number of hours of daylight in summer solstice is $14.89$hours that is,$14$hours and $53$minutes.

Step 3. Number of hours of daylight in the vernal equinox  

(b) To find the number of hours of daylight in the vernal equinox, substitute $i=0°$and $\theta =40.75°$in equation (1)

$\begin{array}{rl}D& =24\left[1-\frac{{\cos }^{-1}(\tan (0)\tan (40.75\left)\right)}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}(0\times 0.86)}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}\left(0\right)}{\pi }\right]\\ & =24\left[1-\frac{\pi }{2\pi }\right]\\ & =24\left[\frac{1}{2}\right]\\ & =12\end{array}$

Therefore, the number of hours of daylight in the vernal equinox is $12$hours.

Step 4. Number of hours of daylight on July 4.

(c) To find the number of hours of daylight on July$4$, substitute the value of $i$and $\theta =40.75°$in equation (1)

Here $i=22°48\text{'}$

We have to convert this into degree, for that divide the minute part by $60$.

localid="1647589248554" $\begin{array}{rl}D& =24\left[1-\frac{{\cos }^{-1}(\tan (22.8)\tan (40.75\left)\right)}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}(0.42\times 0.86)}{\pi }\right]\\ & =24\left[1-\frac{{\cos }^{-1}\left(0.3612\right)}{\pi }\right]\\ & =24\left[1-\frac{\left(0.3821\right)\pi }{\pi }\right]\\ =& 24[1-0.3821]\\ =& 14.83\end{array}$

Therefore, the number of hours of daylight in July is $14.89$hours, that is $14$hours and $49$minutes.