from a lockout tower 25 meters high, a man observes from a position 1.70 meters below the top of the tower that the angle of elevation of the top of a nearby building is 12 degrees 40 minutes and the angle of depression of its base is 70 degrees 20 minutes. If the base of the tower and the buildings are on the same level, what is the height of the building? What is the distance between the tower and the building?
OK. First, it's lookout, not lockout.
The man is 25-1.7 = 23.3m up.
If the building is at distance x from the tower, and has height h, we have
23.3/x = tan 70°20'
(h-23.3)/x = tan 12°40'
Now, you can eliminate x, and then solve for h:
23.3/tan70°20' = (h-23.3)/tan12°40'
To find the height of the nearby building and the distance between the tower and the building, we can use trigonometry and the properties of right-angled triangles. Let's break down the problem step by step.
Let's denote:
- Height of the lockout tower: h1 = 25 meters
- Height of the nearby building: h2 (to be determined)
- Distance between the tower and the building: d (to be determined)
- Angle of elevation: θ1 = 12 degrees 40 minutes
- Angle of depression: θ2 = 70 degrees 20 minutes
1. Finding the height of the building (h2):
From the top of the lockout tower, the man observes the top of the nearby building. This creates a right-angled triangle between the top of the tower, the top of the building, and the base of the building (which is on the same level as the base of the tower).
Using the angle of elevation (θ1) and the height of the man (1.70 meters), we can set up the following equation:
tan(θ1) = (h2 + 1.70) / d
Now, let's substitute the values and solve for h2:
tan(12° 40') = (h2 + 1.70) / d
To solve this equation, convert the angle to decimal degrees:
12° 40' = 12 + 40/60 = 12.66 degrees
tan(12.66°) = (h2 + 1.70) / d
Next, rearrange the equation to solve for h2:
h2 = (tan(12.66°) * d) - 1.70
2. Finding the distance between the tower and the building (d):
From the position 1.70 meters below the top of the tower, the man now observes the base of the nearby building. This creates another right-angled triangle, where the base of the tower, the base of the building, and the top of the tower form the three vertices.
Using the angle of depression (θ2) and the height of the man (1.70 meters), we can write the following equation:
tan(θ2) = (h1 - 1.70) / d
Now, substitute the values and solve for d:
tan(70° 20') = (h1 - 1.70) / d
Convert the angle to decimal:
70° 20' = 70 + 20/60 = 70.33 degrees
tan(70.33°) = (h1 - 1.70) / d
Rearrange the equation to solve for d:
d = (h1 - 1.70) / tan(70.33°)
3. Substitute the values and calculate:
Now that we have both equations, substitute h1 = 25 meters and solve for h2 and d:
h2 = (tan(12.66°) * d) - 1.70
d = (h1 - 1.70) / tan(70.33°)
Using these equations, input the known values (h1 = 25, θ1 = 12.66, θ2 = 70.33) to calculate h2 and d.