You live in a building and your friend lives in the building next to yours. The

Question

You live in a building and your friend lives in the building next to yours. The two of you are having a discussion about the heights of the buildings and your friend claims that his building is half again as tall as yours. To resolve this issue you climb to the roof top of your building and estimate that your line of sight to the top edge of the other building makes an angle of 21° above the horizontal, while your line of sight to the base of the other building makes an angle of 52° below the horizontal. Please determine the ratio of the height of the taller building to the height of the shorter building.

Explanation / Answer

I'm going to leave off the degree sign because it's faster to type.

Let d be the distance between the two buildings.   You are on the roof of the smaller building. The height of the larger building above you is d tan 21. The portion of its height below you is d tan 52.

That, d tan 52, also happens to be the total height of the smaller building since you are on its roof sighting straight across.

So, height of tall building: d(tan 21 + tan 52)

height of short building: d(tan 52)

Their ratio: (tan 21 + tan 52)/tan 52 = (0.38386 + 1.27994)/1.27994 = 1.3

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