Answer:
197 ft
Step-by-step explanation:
You want to know the height of a tower if the angle of elevation to its top is 23° and the angle of depression to its base is 20° from a spot 250 ft away horizontally.
The tangent ratio is ...
Tan = Opposite/Adjacent
In this geometry, the height above the horizontal line of sight is the side of the right triangle that is opposite the angle of elevation. The distance below the line of sight is opposite the angle of depression. The adjacent side is the 250 ft distance between the viewpoint and the tower.
tower height = Adjacent×Tan(23°) +Adjacent×Tan(20°)
tower height = (250 ft)(tan(23°) +tan(20°)) ≈ 197 ft
The tower is about 197 feet tall.
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The height of the tower is approximately 95.64 feet.
To find the height of the tower, we can use trigonometry and the concept of similar triangles. Let's denote the height of the tower as h.
From the window, the person observes the tower's top at an angle of elevation of 23 degrees. This forms a right triangle with the height of the tower as the opposite side and the distance from the window to the tower as the adjacent side. Using the tangent function, we have tan(23°) = h/250.
Similarly, the person observes the bottom of the tower at an angle of depression of 20 degrees. This forms another right triangle with the height of the tower as the adjacent side and the distance from the window to the tower as the opposite side. Using the tangent function, we have tan(20°) = h/250.
Solving these two equations simultaneously, we can find the value of h. By rearranging the equations, we have h = 250 * tan(23°) and h = 250 * tan(20°). Evaluating these expressions using a calculator, we find that h is approximately 95.64 feet.
Therefore, the height of the tower is approximately 95.64 feet.
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