Answer:
- first building: 479.2 ft
- second building: 704.2 ft
Step-by-step explanation:
Label the mailbox viewpoint, the top of the first building, and the top of the second building points M, A, and B, respectively.
Consider the triangle MAB. You are told length MA is 600 ft, and length AB is 800 ft. Side AB is opposite the 67° angle at M, so we can use the Law of Sines to find other measures of that triangle. Specifically, we want to know the distance MB.
sin(B)/MA = sin(M)/AB
sin(B) = (MA/AB)sin(M) = 600/800·sin(67°) ≈ 0.690379
B ≈ 43.66°
Then the angle at A is ...
180° -67° -43.66° = 69.34°
and the side MB can be found from ...
MB/sin(A) = AB/sin(M)
MB = (sin(A)/sin(M))AB = sin(69.34°)/sin(67°)·800 ≈ 813.2 . . . feet
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Now, label points on the ground below A and B as A' and B'. This problem is asking for the heights AA' and BB'. Each of these is opposite the angle of elevation to points A and B, respectively, so can be found using the sine relation:
sin(elevation to A) = AA'/MA
AA' = MA·sin(elevation to A) = 600·sin(53°) ≈ 479.2 . . . ft
Similarly, ...
BB' = MB·sin(elevation to B) = 813.2·sin(60°) ≈ 704.2 . . . feet
The first building is 479.2 ft tall; the second is 704.2 ft tall.
