A SURVEYOR STANDING W25 S OF TOWER MEASURES THE angle of elevation of the top of the tower as 46,5 degrees. From a position E23 S from the tower the elevation of the top is 37,728 degrees. The distance between the two points is 75m, find the height if the tower
Let's denote the height of the tower as h.
From the first position, the surveyor is standing W25 S of the tower and measures an angle of elevation of 46.5 degrees. This can be represented as a right-angled triangle with the tower as the vertical side (h) and the horizontal distance from the surveyor to the base of the tower as the base (25m). We can use trigonometry to find the height of the tower:
tan(46.5) = h/25
h = 25 * tan(46.5)
h ≈ 25 * 1.0635
h ≈ 26.59m
From the second position, the surveyor is standing E23 S from the tower and measures an angle of elevation of 37.728 degrees. This can be represented as another right-angled triangle with the tower as the vertical side (h) and the horizontal distance from the surveyor to the base of the tower as the base (75m). We can use trigonometry to find the height of the tower:
tan(37.728) = h/75
h = 75 * tan(37.728)
h ≈ 75 * 0.766
h ≈ 57.45m
Since the height of the tower should be the same regardless of the position of the surveyor, we can average these two values to find the height of the tower:
Average height = (26.59 + 57.45) / 2
Average height ≈ 83.04 / 2
Average height ≈ 41.52m
Therefore, the height of the tower is approximately 41.52 meters.
To find the height of the tower, we can use trigonometry and the given information.
Let's consider the triangle formed by the surveyor at point W, the top of the tower, and the base of the tower.
In this triangle, the angle of elevation from point W is 46.5 degrees.
We can use the tangent function to find the height of the tower from point W:
tan(46.5°) = height of the tower / distance between the surveyor and the tower's base
Let's calculate the height of the tower from point W:
height of the tower = tan(46.5°) * distance between the surveyor and the tower's base
height of the tower = tan(46.5°) * 75m
height of the tower ≈ 75m * 1.0767
height of the tower ≈ 80.75175m
Now, let's consider the triangle formed by the observer at point E, the top of the tower, and the base of the tower.
In this triangle, the angle of elevation from point E is 37.728 degrees.
We can use the tangent function to find the height of the tower from point E:
tan(37.728°) = height of the tower / distance between the observer and the tower's base
Let's calculate the height of the tower from point E:
height of the tower = tan(37.728°) * distance between the observer and the tower's base
height of the tower = tan(37.728°) * 75m
height of the tower ≈ 75m * 0.7814
height of the tower ≈ 58.605m
Therefore, the height of the tower is approximately 58.605 meters.
To find the height of the tower, we can use trigonometry. Let's label the given points:
- The surveyor at point S (south) at a distance of 25m from the tower.
- The position E (east) at a distance of 23m from the tower.
Step 1: Calculate the length of the base of the triangle (from S to E).
Use the Pythagorean theorem: The square of the hypotenuse (SE) is equal to the sum of the squares of the other two sides.
SE^2 = SB^2 + BE^2
Given:
SB = 25m
BE = 23m
SE^2 = (25m)^2 + (23m)^2
SE^2 = 625m^2 + 529m^2
SE^2 = 1154m^2
Therefore, SE = √1154m ≈ 33.95m (rounded to two decimal places)
Step 2: Calculate the angle at point S using trigonometry.
tan(angle S) = opposite/adjacent
tan(46.5°) = height of the tower/25m
Now, rearrange the equation to solve for the height of the tower:
height of the tower = 25m * tan(46.5°)
height of the tower ≈ 25m * 1.083 ≈ 27.08m (rounded to two decimal places)
Step 3: Calculate the angle at point E using trigonometry.
tan(angle E) = opposite/adjacent
tan(37.728°) = height of the tower/23m
Now, rearrange the equation to solve for the height of the tower:
height of the tower = 23m * tan(37.728°)
height of the tower ≈ 23m * 0.770 ≈ 17.71m (rounded to two decimal places)
Step 4: Average the two height calculations to get a more accurate result.
Average height = (27.08m + 17.71m) / 2
Average height ≈ 44.79m (rounded to two decimal places)
Therefore, the height of the tower is approximately 44.79 meters.