To solve this problem, we need to find the sun's elevation angle above the horizon. The given information includes:

• Height of the building: $500.0 \, \text{m}$
• Length of the shadow: $800.0 \, \text{m}$

We can use the trigonometric function tangent (tan) to relate the height of the building to the length of the shadow. The tangent of the sun's elevation angle $\theta$ is given by:

$\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\text{height of the building}}{\text{length of the shadow}}$

Substituting the values:

$\tan(\theta) = \frac{500.0}{800.0}$

Now, let's calculate the angle $\theta$ by taking the arctangent (inverse tangent) of the ratio:

$\theta = \arctan \left( \frac{500.0}{800.0} \right)$The sun's elevation angle above the horizon is approximately $32.0^\circ$.

Would you like more details, or do you have any further questions? Here are a few related questions that might interest you:

1. How would the elevation angle change if the building were taller or the shadow shorter?
2. Can you calculate the length of the shadow if the elevation angle of the sun were different?
3. What is the relationship between the time of day and the length of shadows cast by objects?
4. How do different factors, like atmospheric refraction, affect the calculation of the sun's elevation angle?
5. Can we use this method to find the height of an object if the shadow length and elevation angle are known?

Tip: In trigonometry, the tangent function is often used to relate the height and distance of objects in problems involving right triangles.