Alyosha and Ivan are standing between two buildings they know to be equal height. The buildings are 500 feet apart. Looking up at the westernmost building, they form a 30 degree angle. Looking to the northeastern building, they form a 45 degree angle.

A) Ivan guesses they are halfway between the buildings. Why is this a bad guess?

B) How far away from each building are they?

Refer to the diagram shown below.

h = height of each building
x = distance from the left building
y = distance from the right building

Because the distance between the buildings is 500 feet, therefore
x + y = 500 (1)

By definition,
tan 45° = h/y = 1
y = h (2)

Also,
tan 30° = h/x = 1/√3
x = h√3 (3)

Part (a)
If Alyosha and Ivan were halfway between the building, then x = y.
Because x ≠ y, they are not halfway between the buildings.

Part (b)
From (1), (2) and (3), obtain
h + h√3 = 500
h(1+√3) = 500
2.732h = 500
h = 183.0 ft

Therefore
x = 183.0 ft
y = 500 - x = 317.0 ft

Answer:
Alyosha and Ivan are 183 feet from the left building, and 317 feet from the right building.