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We have to find the number of hours of daylight in Houston, Texas $29掳45\text{'}$north latitude in the Summer solstice$(i=23.5掳)$,Vernal equinox $(i=0掳)$and on July $4$$(i=22掳48\text{'})$

(a) To find the number of hours of daylight we have to use the formula $D=24\left[1鈭�\frac{{\cos }^{鈭�1}(\tan i鈰�\tan 胃)}{蟺}\right]...........\left(1\right)$

Substitute $i=23.5掳$and $29掳45\text{'}$in this equation,

$\begin{array}{rl}D& =24\left[1鈭�\frac{{\cos }^{鈭�1}(\tan {23.5}^{鈭�}鈰�\tan {29}^{鈭�}{45}^{\mathrm{鈥�}})}{蟺}\right]\\ & =24\left[1鈭�\frac{{\cos }^{鈭�1}\left(0.2485\right)}{蟺}\right]\\ & =24\left[1鈭�\frac{75.61}{蟺}\right].........\left(2\right)\end{array}$

We have to convert $75.61掳$into radians by multiplying by$\frac{\mathrm{蟺}}{180}$.

That is,

$75.61\left(\frac{蟺}{180}\right)=1.319$

Substitute this in equation (2) we get,

$\begin{array}{rl}D& =24[1鈭�\frac{1.319}{蟺}]\\ & =24[1鈭�0.42]\\ & =24\left[0.58\right]\\ & =13.92\\ & ={13}^{鈭�}{55}^{\mathrm{鈥�}}\end{array}$

Therefore, the number of hours of daylight in summer solstice is $13$hours and $55$minutes.

(b) To find the number of hours of daylight in the vernal equinox, substitute $i=0掳$and$胃=29掳45\text{'}$in equation (1)

localid="1647584370028" $\begin{array}{rl}D& =24\left[1鈭�\frac{{\cos }^{鈭�1}(\tan 0^{鈭�}鈰�\tan {29}^{鈭�}{45}^{\mathrm{鈥�}})}{蟺}\right]\\ & =24\left[1鈭�\frac{{\cos }^{鈭�1}\left(0\right)}{蟺}\right]\\ & =24\left[1鈭�\frac{90}{蟺}\right].............\left(3\right)\end{array}$

We have to convert localid="1647584164050" $90掳$into radians by multiplying by$\frac{\mathrm{蟺}}{180}$.

$90\left(\frac{蟺}{180}\right)=1.57$

Substitute this in equation (3) we get,

$\begin{array}{rl}D& =24[1鈭�\frac{1.57}{蟺}]\\ & =24[1鈭�0.5]\\ & =24\left[0.5\right]\\ & =12\end{array}$

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

(c) To find the number of hours of daylight on July$4$, substitute$i=22掳48\text{'}$and $胃=29掳45\text{'}$in equation (1)

$\begin{array}{rl}D& =24\left[1鈭�\frac{{\cos }^{鈭�1}(\tan {22}^{鈭�}{48}^{\mathrm{鈥�}}鈰�\tan {29}^{鈭�}{45}^{\mathrm{鈥�}})}{蟺}\right]\\ & =24\left[1鈭�\frac{{\cos }^{鈭�1}\left(0.24\right)}{蟺}\right]\\ & =24\left[1鈭�\frac{76.113}{蟺}\right]\end{array}$

We have to convert $76.113$degree into radians by multiplying by$\frac{\mathrm{蟺}}{180}$.

localid="1647585959116" $76.113\left(\frac{蟺}{180}\right)=1.328$

Substitute this in equation (4) we get,

$\begin{array}{rl}D& =24[1鈭�\frac{{\cos }^{鈭�1}鈦�\left(0.24\right)}{蟺}]\\ & =24[1鈭�\frac{1.328}{蟺}]\\ & =24[1鈭�0.423]\\ & =24\left[0.577\right]\\ & =13.85\\ & ={13}^{鈭�}{51}^{\mathrm{鈥�}}\end{array}$

Therefore, the number of hours of daylight in July$4$is$13$hours and$51$minutes.