A mailbox on the street is 600 feet from the top of a building. From the mailbox looking up to the tops of that building and a second, taller building, the angle between the buildings is 67degrees. The tops of the buildings are 800 feet from each other. If the angle of elevation from the mailbox to the first building is 53 degrees and the taller building is 60 degrees, how tall is each building.

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.

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step-by-step explanation: