A 60-foot flagpole stands on top of a building. From a point on the ground the angle of elevation to the top of the pole is 45 degrees and the angle of elevation to the bottom of the pole is 42 degrees. How high is the building?
a ladder 6.5 long rests against a wall 4.8 m from the ground.How far from the wall is the foot of the ladder?
To find the height of the building, we need to find the length of the pole first.
Let's assume that the height of the building is h feet and the length of the pole is p feet.
From the given information, we can form a right-angled triangle:
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| \
p | \ h
| \
| \
____________|____\
42° 45°
In this triangle, we have two angles and one side length. We can use trigonometric functions to solve for the unknown sides.
Using the angle of elevation of 45 degrees, we can write:
tan(45°) = p/h
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is p (length of the pole) and the adjacent side is h (height of the building).
tan(45°) = p/h
Since tan(45°) is equal to 1:
1 = p/h
Therefore, p = h (eq. 1)
Now, using the angle of elevation of 42 degrees, we can write:
tan(42°) = (p + 60)/h
The opposite side is p + 60 (length of the pole plus the height of the flagpole), and the adjacent side is still h (height of the building).
tan(42°) = (p + 60)/h
Rearranging this equation, we get:
h = (p + 60) / tan(42°) (eq. 2)
Substituting eq. 1 into eq. 2, we get:
h = (h + 60) / tan(42°)
Multiplying both sides of the equation by tan(42°), we get:
h * tan(42°) = h + 60
Expanding the equation, we get:
h * tan(42°) - h = 60
Factoring out h, we get:
h * (tan(42°) - 1) = 60
Dividing both sides of the equation by (tan(42°) - 1), we get:
h = 60 / (tan(42°) - 1)
Calculating this expression, we find that the height of the building is approximately 89.46 feet.
Therefore, the building is approximately 89.46 feet tall.