Question

The shadow of a building is 30m long when the angle of elevation of the sun is 56°. Calculate the:

(a) height of the building
(b) length of shadow when the sun is at an angle of 32°. ​

Answer 1

(a) height: 44.5 m

(b) shadow: 71.2 m

Explanation:

You want building height and shadow length for angles of elevation 56° and 32°, given the shadow is 30 m when elevation is 56°.

Trig relation

The building height and shadow length are the sides opposite and adjacent (respectively) to the elevation angle in the right triangle that models the geometry. The trig function that relates these sides to the angle is the tangent function:

Tangent = Opposite/Adjacent

Application

(a) building height

Using the tangent relation, we have ...

tan(56°) = (height)/(30 m)

height = (30 m)(tan(56°)) ≈ 44.4768 m

The height of the building is about 44.5 meters.

(b) shadow length

Using the same relation with the building height known, we have ...

tan(32°) = (44.4768 m)/(shadow)

shadow = (44.4768 m)/tan(32°) ≈ 71.1778 m

The length of the shadow at an angle of 32° is about 71.2 m.