Question

Find the height of the tower if the angles of elevation of the top of a tower from two points at a distance of 3 m and 27 m from the base of the tower and in the same straight line with it are complementary.

solve by process ​

Answer 1

Let the height of the tower be CD. B is a point 4 m away from the base C of the tower and A is a point 5 m away from point B in the same straight line. The angles of elevation of the top D of the tower from points B and A are complementary.

Since the angles are complementary if one angle is θ then the other is (90° - θ).

Using tan θ and tan (90° - θ) = cot θ ratios are equated to find the height of the tower.

In ΔBCD,

tan θ = CD/BC

tanθ = CD/4 ....(1)

Here, AC = AB + BC = 5 + 4 = 9

In ΔACD,

tan (90 - θ) = CD/AC

cot θ = CD/9 [Since tan (90- θ) = cot θ]

1/tanθ = CD/9 [As we know that cot θ = 1/tan θ]

tanθ = 9/CD ....(2)

From equation (1) and (2)

CD/4 = 9/CD

CD2 = 36

CD = ± 6

Since height cannot be negative, therefore, the height of the tower is 6 m.

Hence proved that the height of the tower is 6 m.

Answer 2

Answer:

In triangle ABC,

tan C = AB / BC.

tan 30° = AB / 8.

1/√3 = AB / 8

• AB = 8 / √3.

sin C = AB / AC.

sin 30° = (8/√3) / AC.

1/2 = 8/√3 × 1 / AC.

AC = 8/√3 × 2.

• AC = 16 / √3.

Height of tree = AB + AC.

= 8/√3 + 16/√3.

= 24/√3.

= 24 × √3 / √3 × √3.

= (24√3) / 3.

• = 8√3.

Hence, the height of tree is 8√3 meters.