The angle of elevation from the top of a small building to the top of a nearby taller building is 46°40′, while the angle of depression to the bottom is 14°10′. If the shorter building is 28.0 m high, which of the following equations solves for the distance between the buildings? Let x be the distance between the buildings.
Question
The angle of elevation from the top of a small building to the top of a nearby taller building is 46°40′, while the angle of depression to the bottom is 14°10′. If the shorter building is 28.0 m high, which of the following equations solves for the distance between the buildings? Let x be the distance between the buildings.
Solution
The problem involves two right triangles: one formed by the shorter building, the distance between the buildings, and the line of sight to the top of the taller building; and the other formed by the shorter building, the distance between the buildings, and the line of sight to the bottom of the taller building.
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For the first triangle, we can use the tangent of the angle of elevation (46°40′) which is the ratio of the height of the taller building (let's call it h) minus the height of the shorter building (28.0 m) to the distance between the buildings (x). This gives us the equation:
tan(46°40′) = (h - 28.0 m) / x
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For the second triangle, we can use the tangent of the angle of depression (14°10′) which is the ratio of the height of the shorter building (28.0 m) to the distance between the buildings (x). This gives us the equation:
tan(14°10′) = 28.0 m / x
These are the two equations that can be used to solve for the distance between the buildings (x) and the height of the taller building (h).
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