(9)
To find the minimum depth of an overhang that provides 94% shading on July 14th and 11% shading on December 1st for a window located at 31 degrees North latitude, you can use solar angle calculations. The solar angle at solar noon (when the sun is at its highest point in the sky) is crucial for this calculation.

First, you need to find the solar angles for both dates. The solar angle can be calculated using the following formula:

$$
\sin \alpha=\cos \Phi=\sin L \sin \delta+\cos L \cos \delta \cos h
$$

Where,
- Latitude (L) = 31 degrees North
- Solar Declination ($\delta$)= Declination for the given date
- Hour Angle = 0 at solar noon, positive in the morning, and negative in the afternoon
We have to find the solar angle ($\alpha$).[1]

1. To calculate the Solar Declination for July 14th and December 1st we will use

$$
\delta=23.45 \sin \left[\frac{360}{365}(284+N)\right],
$$
where $N$ is the number of days[2].

For July 14th (Summer Solstice):
- Solar Declination (δ) = 23.45° * sin(360° * (284 + 194) / 365) = 21, 8255°

For December 1st (Winter Solstice):
- Solar Declination (δ) = 23.45° * sin(360° * (284 + 334) / 365) ≈ -21, 97°

2. Calculate the Solar Angles at Solar Noon for both dates:

For July 14th:
- Solar Angle$(\alpha)$ = arcsin(sin(31°) * sin(21.83°) + cos(31°) * cos(21.83°) * cos(0)) ≈ 80.83°

For December 1st:
- Solar Angle = arcsin(sin(31°) * sin(-21, 97°) + cos(31°) * cos(-21, 97°) * cos(0)) ≈ 37.03°

3. Now, calculate the overhang depth (d) using the solar angles and desired shading percentages. We can use similar triangles to find the depth:

The formula to find the overhang depth (d) to achieve a certain shading percentage is as follows:

$$ d = \frac{h}{\tan(\text{Solar Angle})} \times \frac{1 - \text{Shading Percentage}}{\text{Shading Percentage}} $$

Where:
- $d$ is the overhang depth.
- $h$ is the height of the window (in this case, 5 ft).
- $\text{Solar Angle}$ is the solar angle calculated for the specific date and time.
- $\text{Shading Percentage}$ is the desired shading percentage (0.94 for 94% and 0.11 for 11%).

For July 14th:
$$ d = \frac{5}{\tan(30.6°)} \times \frac{1 - 0.94}{0.94} \approx 0, 052\text{ ft} .$$

For December 1st:
$$ d = \frac{5}{\tan(37.03°)} \times \frac{1 - 0.11}{0.11} \approx 53, 63 \text{ ft} .$$

So, the minimum overhang depth required to provide 94% shading on July 14th and 11% shading on December 1st for a 5ft window at 31 degrees North latitude is approximately 0.052 feet for July 14th and 53.63 feet for December 1st.

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(10)

The solar hour angle (H) can be used to find the time of sunset. At solar noon, H is 0, and it changes by 15 degrees per hour. To find the time of sunset, we'll calculate H and convert it to hours after local noon.

First, find the solar declination angle (δ) for August 24:

$$δ = -23.45° \cdot \cos\left(\frac{360°}{365} \cdot (N - 172)\right)$$

Where N is the day number. For August 24, N = 236.

Calculate δ:

$$δ ≈ -23.45° \cdot \cos\left(\frac{360°}{365} \cdot (236 - 172)\right) ≈ -10.601°$$

Now, calculate the solar hour angle (H) for sunset:

$$H = \arccos\left(-\tan(34°) \cdot \tan(-10.601°)\right)$$

Calculate H:

$$H ≈ \arccos\left(-\tan(34°) \cdot \tan(-10.601°)\right) ≈ \arccos(0.5547) ≈ 82.747°$$

Now, convert H to hours after local noon:

Since there are 15 degrees per hour, and H is 85.201 degrees:

$$\text{Time after local noon} = H / 15 ≈82.747° / 15 ≈ 5.52 \text{ hours}.$$

So, the local sunset time on August 24 at a latitude of 34°N is approximately 3.77 hours after local noon, which is equivalent to 3 hours and 46.2 minutes after local noon.

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(11)
Solar noon occurs when the sun is at its highest point in the sky. To find the time of solar noon, we need to calculate the solar hour angle (H) at solar noon.

First, find the solar declination angle (δ) for August 27:

$$δ = -23.45° \cdot \cos\left(\frac{360°}{365} \cdot (N - 172)\right)$$

Where N is the day number. For August 27, N = 239.

Calculate δ:

$$δ ≈ -23.45° \cdot \cos\left(\frac{360°}{365} \cdot (239 - 172)\right) ≈ -23.45° \cdot \cos(60.08°) ≈ -9, 51°$$

Now, calculate the solar hour angle (H) at solar noon:

$$H = 0°$$

Solar noon occurs when H is 0 degrees.

Now, calculate how many minutes before or after noon solar noon occurs:

Since solar noon occurs at 12:00 (noon), there are 60 minutes in an hour:

$$\text{Solar noon time} = 12:00$$

So, solar noon on August 27 at the prime meridian occurs exactly at 12:00 (noon).

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(12)
The sol-air temperature represents the temperature that a flat roof surface would experience due to the sun's heating effects. It's typically higher than the ambient temperature. To calculate the sol-air temperature, you can use the following formula:

Sol-Air Temperature = Ambient Temperature + (Solar Gain / (Roof Heat Capacity * Roof Area))

- Ambient Temperature = 48°F
- Solar Gain is the amount of heat gained from the sun, which depends on solar radiation and other factors.
- Roof Heat Capacity is the heat capacity of the roof material.
- Roof Area is the area of the roof.

To calculate Solar Gain, you would need to know the solar radiation and other climate data for Pullman on September 16. This data can vary widely depending on location and weather conditions. You might need to consult local climate records or use weather modeling software to estimate Solar Gain accurately.

Once you have the Solar Gain, Roof Heat Capacity, and Roof Area, you can plug these values into the formula to calculate the Sol-Air Temperature. Without these specific values, it's not possible to provide an exact Sol-Air Temperature for the given scenario.