There are two houses of the same height on both sides of a 40-meter wide road. From a point on the road, elevation angles of the houses are 30° and 60° respectively. Find the height of the houses.

Solution

Let y be the height of the houses, here AB = CD = y meter. Let x be the distance between the point on the road and the house making an elevation angle of 60°, here BE = x meter. Then, (40 - x) is the distance between the point on the road from the house making an elevation angle 30°, here DE = (40 - x) meter. Now, tan 60° = AB/BE ⇒ √3 = y/x ⇒ x = y/√3   ...(i) Also, tan 30° = CD/DE ⇒ 1/√3 = y/(40 - x) ⇒ √3y = 40 - x ⇒ √3y = 40 - y/√3   [from equation i] ⇒ 3y = 40√3 - y ⇒ 4y = 40√3 ⇒ y = 40√3/4 ≈ 17.3 meter