You live in the building on the left in the drawing, and a friend lives in the o

Question

You live in the building on the left in the drawing, and a friend lives in the other building. 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 the issue you climb to the roof 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.

Explanation / Answer

the distance between the buildings = d the vertical part of the upper triangle = h1 (which is the extra height the second building has) the vertical part of the lower triangle = h2 (which is the height of your building) for the top triangle, tan(21) = h1/d for the lower triangle tan(52) = h2/d so tan(21)/tan(52) = h1/h2 = 0.3 the ratio of building heights is (h1+h2)/h2 where h1=h2*0.3 ratio = (h2*0.3 + h2)/h2 = (0.3+1)/1 = 1+0.3 ratio = 1.3 he is wrong, the ratio would need to be 1.5 for him to be right.

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