Question

URGENT!!

1. A 500-foot tall monument is located in the distance. From a window in a building, person determines that the angle of elevation to the top of the monument is 13°, and that the angle of depression to the bottom of the monument is 2°. How far is the person from the monument? Round your answer to two decimal places.
2. The angle of elevation to the top of a building in New York is found to be 3 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. Round your answer to four decimal places.

Answer 1

1. 1881.19 feet
2. 0.0524 miles or 276.7131 feet

Explanation:

You want the distance to a 500 ft monument given an angle of elevation of 13° to its top and -2° to its bottom. You also want the height of a building if the angle of elevation from 1 mile distant is 3°.

1. Monument

The ratio of opposite side to adjacent side in a right triangle is the tangent of the angle:

Tan = Opposite/Adjacent

If we let the distance to the monument be represented by 'd', then the distance to the ground from the observation height is ...

tan(2°) = (distance to ground)/d

Similarly, the distance to the top from the observation height is ...

tan(13°) = (distance to top)/d

The sum of the distance to the ground and the distance to the top is the height of the monument:

d·tan(2°) +d·tan(13°) = monument height = 500 ft

Then the distance to the monument is ...

d = (500 ft)/(tan(2°) +tan(13°)) ≈ 1881.19 ft . . . distance to the monument

2. Building

The ratio of the building height to the distance from the observation point is the tangent of the angle, as above.

tan(3°) = height/(1 mile)

height = (1 mile)·tan(3°) = 0.0524 mile

There are 5280 feet in a mile, so this is ...

0.0524 mi = (5280 ft)(0.0524) = 276.7131 ft

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Additional comment

The best angle measurement has an accuracy of perhaps 30 microradians, an arc length of about 2 inches at 1 mile. Reporting the building height to the nearest 0.0001 foot, or 0.0012 inches is a claim of more accuracy than is likely in the measurement of the building height. 0.0001 miles is about 6 inches, so a 4 dp value in miles is reasonably justified.