The angle of elevation to the top of a Building in New York is found to be 3 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building to 2 decimal places.

To find the height of the building, we can use trigonometry, specifically the tangent function, which relates the angle of elevation to the height of the building and the distance from the base.

Given:
- Angle of elevation, \(\theta = 3^\circ\)
- Distance from the base of the building, \(d = 2\) miles

The tangent of the angle of elevation is given by:

\[
\tan(\theta) = \frac{\text{height of the building (h)}}{\text{distance from the building (d)}}
\]

So, the height \(h\) can be calculated as:

\[
h = d \times \tan(\theta)
\]

Substituting the given values:

\[
h = 2 \times \tan(3^\circ)
\]

Let me calculate the height.The height of the building is approximately **553.43 feet** to two decimal places.

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How does the angle of elevation change as the observer moves closer to the building?
2. What would be the height of the building if the angle of elevation was 5 degrees instead of 3 degrees?
3. How would you calculate the distance from the building if the height and angle of elevation were known?
4. What other trigonometric functions could be useful in solving problems involving angles of elevation or depression?
5. How can we apply this method to find the height of natural structures like mountains?

**Tip:** When dealing with very small angles, such as 3 degrees, the tangent function produces small values, making it important to ensure calculations are precise to avoid significant errors.

Learn how to find the height of a building using trigonometry with a given angle of elevation and distance from the base. This problem involves using the tangent function to solve for the building's height. Suitable for high school students in Grades 10-12.

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