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To solve this problem, we will use trigonometry. Specifically, we can use the tangent function, which relates the angle of elevation \(\theta\), the height of the building, and the length of the shadow.

The formula for the tangent of the angle is:

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

Here:
- The height of the building is 82.39 feet.
- The length of the shadow is 85.23 feet.

Thus, we can write:

\[
\tan(\theta) = \frac{82.39}{85.23}
\]

Now, let's calculate \(\theta\) by taking the inverse tangent (\(\tan^{-1}\)) of the ratio.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, you can calculate the angle of elevation using the following steps:

1. Divide the height of the building by the length of the shadow:
   \[
   \frac{82.39}{85.23} \approx 0.967
   \]
   
2. Take the inverse tangent (arctan) of the result to get the angle in radians:
   \[
   \theta = \tan^{-1}(0.967)
   \]

3. Convert the angle from radians to degrees by multiplying by \( \frac{180}{\pi} \).

4. Round the result to the nearest hundredth.

Would you like any further clarification or help with these steps?

Here are 5 related questions for you to explore:
1. What is the sine and cosine of the angle you just found?
2. How would the angle of elevation change if the building were taller?
3. How is trigonometry used in real-world applications like surveying or architecture?
4. What is the relationship between the angle of elevation and the length of the shadow?
5. How do inverse trigonometric functions help in finding angles?

**Tip:** Always check that your calculator is in degree mode when working with angles in degrees.

A building 82.39 feet tall has a shadow that is 85.23 feet long. Find the angle of elevation of the sun to the nearest hundredth of a degree.

Calculate the angle of elevation of the sun for a building with a height of 82.39 feet and a shadow length of 85.23 feet using trigonometric functions. Learn how to apply the tangent formula to solve for angles.

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