From a certain distance, you observe the Leaning Tower of Pisa at an angle of elevation of 30 degrees. You walk 100 meters closer to the tower and now observe it at a 45-degree angle of elevation. Calculate the height of the tower.
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Let's denote the height of the tower as "h" and the distance from the observer to the tower as "x".
From the given information, we can set up the following right triangles:
Triangle 1:
Opposite side = h
Adjacent side = x
Angle of elevation = 30 degrees
Triangle 2:
Opposite side = h
Adjacent side = x - 100
Angle of elevation = 45 degrees
Using the tangent function in both triangles, we have:
In Triangle 1: tan(30 degrees) = h / x
In Triangle 2: tan(45 degrees) = h / (x - 100)
We can rewrite the equations as:
1/sqrt(3) = h / x
1 = h / (x - 100)
From the first equation, we can solve for x:
x = h / (1/sqrt(3))
x = sqrt(3) * h
Substituting x in the second equation:
1 = h / (sqrt(3) * h - 100)
Simplifying:
sqrt(3) * h - 100 = h
sqrt(3) * h - h = 100
(h * (sqrt(3) - 1)) = 100
h = 100 / (sqrt(3) - 1)
Using a calculator, we can compute:
h ≈ 63.66 meters
So, the height of the Leaning Tower of Pisa is approximately 63.66 meters.
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