**Frequently Asked Questions on Chapter 9 Some Applications of Trigonometry**

**The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.**

Let AB be the height of the tower and C is the point elevation which is 30 m away from the foot of the tower.

To Find: AB (height of the tower)

Let us consider in △ABC

tan 30° = AB/BC

1/√3 = AB/30

AB = 10√3

Hence, the height of the tower is 10√3 m.

**A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.**

Let us consider in △ABC

Let BC = Height of the kite from the ground, BC = 60 m

AC = Inclined length of the string from the ground and A is the point where the string of the kite is tied.

To Find: Length of the string from the ground i.e. the value of AC

So,

sin 60° = BC/AC

√3/2 = 60/AC

AC = 40√3 m

Hence, the length of the string from the ground is 40√3 m.

**From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.**

Let BC be the 20m high building.

D is the point on the ground from where the elevation is taken.

Height of transmission tower = AB = AC – BC

To Find: AB, Height of the tower

Let us consider in right ΔBCD,

tan 45° = BC/CD

1 = 20/CD

CD = 20

Again,

In right ΔACD,

tan 60° = AC/CD

√3 = AC/20

AC = 20√3

Now, AB = AC – BC = (20√3 – 20) = 20(√3 – 1)

Hence, the height of the transmission tower is 20(√3 – 1) m.

In right ΔABC,

tan x = AB/BC

tan x = AB/4

AB = 4 tan x … (i)

Again, from right ΔABD,

tan (90°-x) = AB/BD

cot x = AB/9

AB = 9 cot x … (ii)

Multiplying equation (i) and (ii)

AB^{2} = 9 cot x × 4 tan x

⇒ AB^{2} = 36 (because cot x = 1/tan x

⇒ AB = ± 6

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

Hence Proved.