Solution of Chapter 9. Some Applications of Trigonometry (Maths Important Questions Book)

Important Questions

1
From a point 20 m away from the foot of a tower, the angle of elevation of thetop of the tower is 30 ยบ. Find the height of the tower.
2
A ladder is placed against a wall such that it just reaches the top of thewall. The foot of the ladder is 1.5 m away from the wall and the ladder isinclined at an angle of 60 owith the ground. Findthe height of the wall.
3
An electric pole is 10 m high. A steel wire tied to the top of the pole isaffixed at a point on the ground to keep the pole upright. If the wiremakes an angle of 45º with the horizontal through the foot of the pole,find the length of the wire.
4

A balloon is connected to a meteorological ground station by a cable of length215 m inclined at 60 º to the horizontal. Determinethe height of the balloon from the ground. Assume that there isno stack in the cable.

5
A bridge across a river makes an angle of 45º with the river bank(fig.). If the length of the bridge across the river is 150 m, what is thewidth of the river?
https://gs-post-images.s3.amazonaws.com/user_files/63094/15047/images/extra-9_files/Image6.gif
6

The angle of elevation of the top of a hill at the foot of a tower is 60ºand the angle of elevation of top of the tower from the foot of the hillis 30º. If the tower is 50 m high, what is the height of the hill?

7

There is a small island in the middle of a 100 m wide river and atall tree stands on the island. P and Q are points directly opposite eachother on the two banks, and in line with the tree. If the anglesof elevation of the top of the tree from P and Q are respectively 30º and45º, find the height of the tree.

8
A flag-staff stands at the top of a 5 m high tower. From a point on the ground,the angle of elevation of the top of the flag-staff is 60º and from the samepoint, the angle of elevation of the top of the tower is 45º. Find the heightof the flag-staff.
9

A vertical tower stands on a horizontal plane and is surmounted by aflag-staff of height 7 m. From a point on the plane, the angle ofelevation of the bottom of the flag-staff is 30º and that of the top ofthe flag-staff is 45º. Find the height of the tower.

10
Determine the height of a mountain if the elevation of its top at an unknowndistance from the base is 25 o10' and at a distance of 5km further off from the mountain, along the same line, the angleof elevation is 15 o20'. Give the answer in km correct to2 decimals.
11

The angle of elevation of a cliff from a fixed point A is θ. After going up adistance of k metres towards the top of the cliff at an angle φ, itis found that the angle of elevation is α. Show that the height ofthe cliff in metre is https://gs-post-images.s3.amazonaws.com/user_files/63094/15047/images/extra-9_files/11d20.gif .

12
The length of a string between a kite and a point on the roof of the building10 m high is 180 m. If the string makes an angle θ with the level ground suchthat tan θ = 4/3 how high is the kite from the ground?
13
If the angle of elevation of a cloud from a point h metres above a lake is αand the angle of depression of its reflection in the lake is β , provethat the height of the cloud ishttps://gs-post-images.s3.amazonaws.com/user_files/63094/15047/images/extra-9_files/Image405.gif .
14
An aeroplane flying horizontally at height of 2500 √ 3 mtsabove that ground; is observed to be at angle of elevation 60° from theground. After a flight of 25 seconds the angle of elevation is 30° . Findthe speed of the plane in m/sec.
15
In the figure below, some dimensions of a hut have been marked. Findthe other dimensions (marked ?) of the hut correctly up to one placeof decimal.
16
A ladder rests against a wall at an angle α to the horizontal. When its foot ispulled away from the wall through a distance a, it slides a distance b down thewall and makes an angle β with the horizontal. Show that a/b = (cosα - cosβ)/(sinα - sinβ ).
17
The line joining the top of a hill to the foot of the hill makes an angle of30° with the horizontal through the foot of the hill. There is onetemple at the top of the hill and a guest house half way from the foot tothe hill . The top of the temple and the top of the guesthouse both make anelevation of 32° at the foot of the hill. If the guesthouse is 1 kmaway from the foot of the hill along the hill, find the heights of theguest house and the temple.
18
A man standing 'a' metres behind and opposite the middle of a football goalobserves that the angle of elevation of the nearer cross-bar is α and that ofthe further crossbar is β. Show that the length of the field is, a(tanα cotβ +1).
19
The angle of elevation of the top of a tower from a point A due south of thetower is α and from B due east of the tower is β.
If AB = d, show that the height of the tower ishttps://gs-post-images.s3.amazonaws.com/user_files/63094/15047/images/extra-9_files/Image2822.gif .
20
A tower subtends an angle α at a point A in the place of its base and the angleof depression of the foot of the tower at a point b ft. just above A is β.Prove that the height of the tower is b tanα cotβ.