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A mine shaft with a circular entrance has been carved into the side of a mountain. From a distance of 350
feet from the base of the mountain, the angle of elevation to the bottom of the circular opening is
The angle of elevation to the top of the opening is . Determine the diameter of the circular entrance.

Ask by Owen Ray. in the United States
Mar 28,2025

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The diameter of the circular entrance is approximately 24.5 feet.

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Beyond the Answer

To find the diameter of the circular entrance from the given angles of elevation, we can model the scenario with a right triangle.
  1. Calculating the height of the bottom and top of the opening:
    Let be the height of the bottom of the opening, and be the height of the top of the opening.
    Using the angle of elevation to the bottom:
    Using the angle of elevation to the top:
  2. Calculating heights:
    • First, calculate and :
  3. Finding the height of the circular entrance:
    The height of the circular entrance, , is the difference between and :
  4. Finding the diameter of the entrance:
    The diameter is equal to the height of the circular opening. Thus, the diameter is:
Therefore, the diameter of the circular entrance is approximately 31.85 feet.

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