A man standing on the roof of a building 64.0 feet high looks down to the building next door. He finds the angle of depression to the roof of that building from the roof of his building to be 34.7°, while the angle of depression from the roof of his building to the bottom of the building next door is 63.3°. How tall is the building next door? (Round your answer to the nearest tenth.)

The height of the next door building is 41.7 feet

Step-by-step explanation:

* Lets study the situation in the problem

- The man standing on the roof of a building 64.0 feet high

- The angle of depression to roof of the next door building is 34.7°

- The angle of depression to the bottom of the next door building

is 63.3°

- We need to find the height of the next door building

* Lets consider the height of the man building and the horizontal

distance between the two building formed a right triangle and the

angle of depression is opposite to the side which represented the

height of the building

- Let the horizontal distance between the two buildings called x

# In the triangle

∵ The length of the side opposite to the angle of depression (63.3°)

is 64.0

∵ The length of the horizontal distance is x which is adjacent to the

angle of depression (63.3°)

- Use the trigonometry function tanФ = opposite/adjacent

∴ tan 63.3° = 64.0/x ⇒ use cross multiplication

∴ x (tan 63.3°) = 64 ⇒ divide both sides by (tan 63.3°)

∴ x = 64.0 / (tan 63.3°)

∴ x = 32.1886 feet

- Lets use this horizontal distance to find the vertical distance between

the roofs of the two buildings

* Lets consider the height of the vertical distance between the roofs

of the two buildings and the horizontal distance between the two

building formed a right triangle and the

angle of depression is opposite to the side which represented the

vertical distance between the roofs of the two buildings

- Let the vertical distance between the roofs of the two buildings

called y

# In the triangle

∵ The vertical distance between the roofs of the two buildings is y

and opposite to the angle of depression (34.7°)

∵ The horizontal distance x is adjacent to the angle of

depression (34.7°)

∴ tan (34.7°) = y/x

∵ x = 32.1886

∴ tan 34.7° = y/32.1886 ⇒ use the cross multiplication

∴ y = 32.1886 (tan 34.7°)

∴ y = 22.2884 ≅ 22.3 feet

∴ The vertical distance between the roofs of the two

buildings is 22.3 feet

- The height of the next door building is the difference between the

height of the man building and the vertical distance between the

roofs of the two buildings

∴ The height of the next door building = 64.0 - 22.3 = 41.7 feet