Answer: The height of the first building is 479.16 feet, while the height of the taller building is 704.24 feet.
Step-by-step Explanation: Please refer to the attached diagram for details. Let both buildings be A and B. The tops of both buildings are 800 feet from each other which is line AB. The mailbox is 600 feet from the top of building A which is line AM. In triangle ADM, AD the height of the first is calculated as,
Sin 53 = opposite/hypotenuse
Sin 53 = AD/600
0.7986 = AD/600
0.7986 x 600 = AD
479.16 = AD
Then looking at triangle ABM,
Applying the sine rule, we now have
800/Sin67 = 600/SinB
By cross multiplication we now have
SinB = (600 x Sin67)/800
SinB = (600 x 0.9205)/800
SinB = 552.3/800
SinB = 0.690375
B = 43.66 degrees
With that we can determine angle to be;
Angle A = 180 - (67 + 43.66)
Angle A = 180 - 110.66
Angle A = 69.34
Therefore to calculate line BM using the sine rule,
BM/Sin69.34 = 800/Sin67
By cross multiplication we now have
BM = (800 x Sin69.34)/Sin67
BM = (800 x 0.9357)/0.9205
BM = 748.56/0.9205
BM = 813.21
From this point we can now calculate the height of the other building which is line BC.
Looking at triangle BCM, line BC can be calculated as follows;
Sin60 = opposite/hypotenuse
Sin60 = BC/813.21
0.866 x 813.21 = BC
704.2398 = BC
Approximately BC = 704.24
Therefore the height of the first building is 479.16 feet while the height of the taller building is 704.24 feet.
