to measure the height of the Eiffel Tower in Paris, a person stands away from the base and measures the angle of elevation to the top of the tower to be 60 degrees. Moving 210 feet closer, the angle of elevation to the top of the tower is 70 degrees. How tall is the Eiffel Tower?
To calculate the height of the Eiffel Tower, we can use trigonometry and the concept of similar triangles. Here's how you can find the height of the tower:
1. Draw a diagram: Start by drawing a diagram to better visualize the situation. Draw a vertical line to represent the Eiffel Tower, and label the top as 'T'. Then, draw a person standing some distance away from the base of the tower, with an angle of elevation of 60 degrees. Mark this point as 'P'.
2. Set up the problem: Let's assume the height of the Eiffel Tower is 'h' feet. From the given information, we know that the person's original distance from the base of the tower is 'd' feet, and the new distance after moving closer is 'd - 210' feet. The angles of elevation are 60 degrees and 70 degrees, respectively.
3. Apply trigonometry: Since we have a right triangle formed by the person, the tower, and a line connecting them, we can use the tangent function to relate the angle of elevation to the height and distance.
In triangle TPO (with right angle at P):
tan(60 degrees) = h / d
In triangle TQO (with right angle at Q, where Q is the new position):
tan(70 degrees) = h / (d - 210)
4. Solve the equations: We now have a system of two equations with two unknowns, h and d. Since we need to find the height of the tower, we can solve for h.
From equation 1: tan(60 degrees) = h / d
Therefore, h = d * tan(60 degrees)
Plug the expression for h in equation 2, and solve for d:
tan(70 degrees) = (d * tan(60 degrees)) / (d - 210)
Simplify the equation and solve for d.
5. Calculate the height: Once you find the value of d, you can substitute it back into equation 1 to find the height h.
h = d * tan(60 degrees)
By using these calculations, you can determine the height of the Eiffel Tower based on the given angles of elevation and distances.