Calculating the Angle of Elevation When Viewing the Eiffel Tower

The Eiffel Tower, an iconic structure located in Paris, France, stands at a monumental height of 324 meters. If a tourist is looking at it from 100 meters away from the base, how can we calculate the angle at which they are tilting their head to see the top? This calculation involves the application of trigonometry, particularly the tangent function. Let's walk through the steps to find the angle of elevation.

Using Trigonometry to Find the Angle

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In this scenario, we can use the basic trigonometric function, specifically the tangent, to find the angle of elevation.

To start, we need to set up the problem. The Eiffel Tower's height (opposite side) is 324 meters, and the distance from the tourist to the base of the tower (adjacent side) is 100 meters.

The formula for the tangent is given by:

( tan(theta) frac{text{opposite}}{text{adjacent}} )

Plugging in the values, we get:

( tan(theta) frac{324}{100} 3.24 )

So, we need to find the angle ( theta ) for which the tangent value is 3.24. This can be done by taking the inverse tangent (( arctan )) of 3.24:

( theta arctan(3.24) )

Using a calculator to find the value of ( arctan(3.24) ), we get:

( theta approx 73.74^circ )

Therefore, the angle at which the tourist must tilt their head to see the top of the Eiffel Tower is approximately 73.74 degrees.

Further Considerations

It is important to note that the calculation assumes the tourist's eyes are at ground level. In reality, most tourists' eyesight would be slightly above ground level due to their head's height. However, for mathematical simplicity, the approximation of an eye level of 1.7 meters (a typical adult height) could be used:

( 324 - 1.7 322.3 ) meters (vertical side) ( 100 ) meters (horizontal side)

Repeating the calculation with these values:

( tan(theta) frac{322.3}{100} 3.223 )

( theta arctan(3.223) approx 72.85^circ )

This gives us an angle of elevation of approximately 72.85 degrees.

Real-World Implications

Understanding the angle of elevation is crucial in various fields, including architecture, engineering, and astronomy. It allows for the precise measurement and calculation of angles of sight, which is vital for designing and planning structures and observations.

In conclusion, the angle of elevation when viewing the Eiffel Tower from 100 meters away is approximately 72.85 degrees. This calculation not only meets the requirement of a challenging trigonometry problem but also provides practical insights into the application of mathematical principles in real-world scenarios.