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Answer :
Final answer:
To find the diameter of the circular entrance, first, find the distances from the observer to the top and bottom of the opening using trigonometry. Then, calculate the heights using the tangent function with the given angles. Finally, double the height to find the diameter.
Explanation:
To solve this problem, we can use the concept of trigonometry. First, we need to find the distance from the observer to the top and bottom of the circular opening using the given angles of elevation. Then, we can use the tangent function to calculate the height of the circular opening. Finally, we can double the height to find the diameter of the circular entrance.
Let's start by finding the height using the angle of elevation to the bottom of the opening. The tangent of the angle is equal to the opposite side (height) divided by the adjacent side (distance from the observer to the bottom of the opening). Using the formula, we have tan(36.3°) = height / 200. Rearranging the equation, height = tan(36.3°) * 200.
Similarly, we can find the height using the angle of elevation to the top of the opening. The tangent of the angle is equal to the opposite side (height) divided by the adjacent side (distance from the observer to the top of the opening). Using the formula, we have tan(41°) = height / 200. Rearranging the equation, height = tan(41°) * 200.
Now that we have the heights, we can calculate the diameter of the circular entrance by doubling the height. So, the diameter = 2 * height.
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Final answer:
To determine the diameter of the circular entrance, we can use trigonometry and the given angles of elevation.
Explanation:
To determine the diameter of the circular entrance, we can use trigonometry and the given angles of elevation. Let's denote the diameter as 'd'. Since the angle of elevation to the bottom of the opening is 36.3 degrees, we can set up the following trigonometric equation: tan(36.3) = h / 200, where 'h' is the height from the base to the bottom of the opening. Similarly, for the top of the opening, we have tan(41) = (h + d) / 200. Solving these equations simultaneously will give us the value of 'd'.
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