Question

A tower stands 50 meters tall. From its peak, the angles of depression to the top and bottom of a nearby building are measured at 30° and 45°, respectively. Determine the approximate height of the building as well as the horizontal distance separating the tower from the building.

• A. height = 21m , distance = 50m
• B. height = 25m , distance = 25m
• C. height = 28m , distance = 15m
• D. height = 18m , distance = 15m

Answer 1

Answer:

A. height = 21m , distance = 50m

Let the tower height be 50 m.

Let:

• Height of the building = (h) m

• Horizontal distance between tower and building = (d) m

1. Find the horizontal distance

The angle of depression to the bottom of the building is (45^\circ).

$\tan45^\circ=\frac{50}{d}$

Since $(\tan45^\circ=1),$

$d=50\text{ m}$

2. Find the height of the building

The angle of depression to the top of the building is (30^\circ).

The vertical difference between the tower top and building top is:

50-h

Therefore,

$\tan30^\circ=\frac{50-h}{50}$

Since $(\tan30^\circ=\frac1{\sqrt3}),$

$50-h=\frac{50}{\sqrt3}$

$h=50-\frac{50}{\sqrt3}$

$h\approx50-28.87$

$\boxed{h\approx21.13\text{ m}}$

Final Answer

• Height of building = 21m

• Horizontal distance = 50 m