Solution: Consider P as the man standing on the deck of the ship which is 20 m above the sea level and B is the bird Angle of elevation of the bird from P = 300 Angle of depression from P to the...
There is a small island in between a river 100 metres wide. A tall tree stands on the island. P and Q are points directly opposite to each other on the two banks and in the line with the tree. If the angles of elevation of the top of the tree from P and Q are 300 and 450 respectively, find the height of the tree.
Solution: Width of the river PQ = 100 m B is the island and AB is the tree on it Angles of elevation from A to P and Q are 300 and 450 Consider AB = h PB = x BQ = 100 – x In right triangle APB tan θ...
A man on the deck of a ship is 16 m above the water level. He observes that the angle of elevation of the top of a cliff is 450 and the angle of depression of the base is 300. Calculate the distance of the cliff from the ship and the height of the cliff. Solution:
Solution: Consider A as the man on the deck of a ship B and CE is the cliff AB = 16 m Angle of elevation from the top of the cliff = 450 Angle of depression at the base of the cliff = 300 Take CE =...
An aeroplane is flying horizontally 1 km above the ground is observed at an elevation of 600. After 10 seconds, its elevation is observed to be 300. Find the speed of the aeroplane in km/hr.
Solution: Consider A and D as the two positions of the aeroplane AB is the height and P is the point AB = 1 km Take AD = x and PB = y Angles of elevation from A and D at the point P are 600 and 300...
In the adjoining figure, the angle of elevation of the top P of a vertical tower from a point X is 60 degree; at a point Y, 40 m vertically above X, the angle of elevation is 450. Find (i) the height of the tower PQ (ii) the distance XQ (Give your answer to the nearest metre)
Solution: Consider PQ as the tower = h XQ = YR = y XY = 40 m PR = h – 40 In right triangle PXQ tan θ = PQ/XQ Substituting the values tan 600 = h/y So we get √3 = h/y y = h/√3 ….. (1) In right...
A boy 1.54 m tall can just see the sun over a wall 3.64 m high which is 2.1 m away from him. Find the angle of elevation of the sun.
Solution: Consider AB as the boy and CD as the wall which is at a distance of 2.1 m AB = 1.54 m CD = 3.64 m BD = 2.1 m Construct AE parallel to BD ED = 1.54 m CE = 3.64 – 1.54 = 2.1 m AE = BD = 2.1...
A boy 1.6 m tall is 20 m away from a tower and observes that the angle of elevation of the top of the tower is 600. Find the height of the tower. Solution: Consider AB as the boy and TR as the...
A 7 m long flagstaff is fixed on the top of a tower. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 450 and 360 respectively. Find the height of the...
An aeroplane 3000 m high passes vertically above another aeroplane at an instant when the angles of elevation of the two aeroplanes from the same point on the ground are 600 and 450 respectively....
The angle of elevation of the top of a tower from a point A (on the ground) is 300. On walking 50 m towards the tower, the angle of elevation is found to be 600. Calculate (i) the height of the...
An aircraft is flying at a constant height with a speed of 360 km/h. From a point on the ground, the angle of elevation of the aircraft at an instant was observed to be 450. After 20 seconds, the...
In the adjoining figure, the shadow of a vertical tower on the level ground increases by 10 m, when the altitude of the sun changes from 450 to 300. Find the height of the tower and give your answer, correct to 1/10 of a metre.
Solution: Consider TR as the tower where TR = h BR = x AB = 10 m Angles of elevation from the top of the tower at A and B are 300 and 450 In right triangle TAR tan θ = TR/AR Substituting the values...
The angle of elevation of the top of an unfinished tower at a point distant 120 m from its base is 450. How much higher must the tower be raised so that its angle of elevation at the same point may...
From the top of a building 20 m high, the angle of elevation of the top of a monument is 450 and the angle of depression of its foot is 150. Find the height of the monument.
Solution: Consider AB as the building where AB = 20 m CD as the monument where CD = x m Take the distance between the building and the monument as y In right triangle BCD tan θ = CD/BD Substituting...
A vertical pole and a vertical tower are on the same level ground. From the top of the pole the angle of elevation of the top of the tower is 600 and the angle of depression of the foot of the tower is 300. Find the height of the tower if the height of the pole is 20 m.
Solution: Consider TR as the tower PL as the pole on the same level Ground PL = 20 m From the point P construct PQ parallel to LR ∠TPQ = 600 and ∠QPR = 300 Here ∠PRL = ∠QPR = 300 which are the...
A pole of height 5 m is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 600 and the angle of depression of the point A from the...
The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 300 and 450 respectively. Find the height of tire multi-storeyed building and the distance between the two buildings, correct to two decimal places.
Solution: Consider AB as the height and CD as the building The angles of depression from A to C and D are 300 and 450 ∠ACE = 300 and ∠ADB = 450 CD = 8 m Take AB = h and BD = x From the point C...
A man 1.8 m high stands at a distance of 3.6 m from a lamp post and casts a shadow of 5.4 m on the ground. Find the height of the lamp post.
Solution: Consider AB as the lamp post CD is the height of man BD is the distance of man from the foot of the lamp FD is the shadow of man Construct CE parallel to DB Take AB = x and CD = 1.8 m EB =...
From a tower 126 m high, the angles of depression of two rocks which are in a horizontal line through the base of the tower are 160 and 120 20’. Find the distance between the rocks if they are on...
(i) The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 600 and 450 respectively. If the two ships are on the opposite sides of the light house,...
From two points A and B on the same side of a building, the angles of elevation of the top of the building are 300 and 600 respectively. If the height of the building is 10 m, find the distance between A and B correct to two decimal places.
Solution: In triangle DBC tan 600 = 10/BC Substituting the values √3 = 10/BC BC = 10/√3 In triangle DBC tan 300 = 10/ (BC + AB) Substituting the values 1/√3 = 10/[10/√3 + AB] By further calculation...
The angle of elevation of a pillar from a point A on the ground is 450 and from a point B diametrically opposite to A and on the other side of the pillar is 600. Find the height of the pillar, given that the distance between A and B is 15 m.
Solution: Consider CD as the pillar of x m Angles of elevation of points A and B are 450 and 600 It is given that AB = 15 m Take AD = y DB = 15 – y In right triangle CAD tan θ = CD/AD Substituting...
As observed from the top of a 80 m tall light house, the angles of depression of two ships on the same side of the light house in horizontal line with its base are 300 and 400 respectively. Find the...
The horizontal distance between two towers is 140 m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 300. If the height of the second tower is 60 m, find the height of the first tower.
Solution: Consider the height of the first tower TR = x It is given that Height of the second tower PQ = 60 m Distance between the two towers QR = 140 m Construct PL parallel to QR LR = PQ = 60 m PL...
In the adjoining figure, not drawn to the scale, AB is a tower and two objects C and D are located on the ground, on the same side of AB. When observed from the top A of the tower, their angles of depression are 450 and 600. Find the distance between the two objects. If the height of the tower is 300. Give your answer to the nearest metre.
Solution: Consider CB = x and DB = y AB = 300 m In right triangle ACD tan θ = AB/CB Substituting the values tan 450 = 300/x 1 = 300/x So we get x = 300 m In right triangle ADB tan θ = AB/DB...
From the top of a church spire 96 m high, the angles of depression of two vehicles on a road, at the same level as the base of the spire and on the same side of it are x0 and y0, where tan x0 = ¼ and tan y0 = 1/7. Calculate the distance between the vehicles.
Solution: Consider CH as the height of the church A and B are two vehicles which make an angle of depression x0 and y0 from C Take AH = x and BH = y In a right triangle CBH tan x0 = CH/AH = 96/y...
In the figure, not drawn to scale, TF is a tower. The elevation of T from A is x0 where tan x = 2/5 and AF = 200 m. The elevation of T from B, where AB = 80 m, is y0. Calculate: (i) the height of...
At a point on level ground, the angle of elevation of a vertical lower is found to be such that its tangent is 5/12. On walking 192 m towards the tower, the tangent of the angle is found to be ¾. Find the height of the tower.
Solution: Consider TR as the tower and P as the point on the ground such that tan θ = 5/12 tan α = ¾ PQ = 192 m Take TR = x and QR = y In right triangle TQR tan α = TR/QR = x\y So we get 3/4 = x/y y...
A man observes the angles of elevation of the top of a building to be 300. He walks towards it in a horizontal line through its base. On covering 60 m the angle of elevation changes to 600. Find the height of the building correct to the nearest decimal place.
Solution: It is given that AB is a building CD = 60 m In triangle ABC tan 600 = AB/BC √3 = AB/BC So we get BC = AB/√3 ….. (1) In triangle ABD tan 300 = AB/BD 1/√3 = AB/ (BC + 60) By cross...
From the top of a hill, the angles of depression of two consecutive kilometer stones, due east are found to be 300 and 450 respectively. Find the distance of two stones from the foot of the hill.
Solution: Consider A and B as the position of two consecutive kilometre stones Here AB = 1 km = 1000 m Take Distance BC = x m Distance AC = (1000 + x) m In right angled triangle BCD CD/BC = tan 450...
The shadow of a vertical tower on a level ground increases by 10 m when the altitude of the sun changes from 450 to 300. Find the height of the tower, correct to two decimal places.
Solution: In the figure AB is the tower BD and BC are the shadow of the tower in two situations Consider BD = x m and AB = h m In triangle ABD tan 450 = h/x So we get 1 = h/x h = x ….. (1) In...
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 600; when he retires 20 m from the bank, he finds the angle to be 300. Find the height of the tree and the breadth of the river.
Solution: Consider TR as the tree and PR as the width of the river. Take TR = X and PR = y In right triangle TPR tan θ = TR/PR Substituting the values tan 600 = x/y So we get √3 = x/y x = y...
An aeroplane when flying at a heigt of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 300 and 600 respectively. Find the distance between the two planes at the instant
Solution: Consider the distance between two planes = h m It is given that AD = 3125 m, ∠ACB = 600 and ∠ACD = 300 In triangle ACD tan 300 = AD/AC Substituting the values 1/√3 = 3125/AC AC = 3125√3...
From a point P on the ground, the angle of elevation of the top of a 10 m tall building and a helicopter hovering over the top of the building are 300 and 600 respectively. Find the height of the helicopter above the ground.
Solution: Consider AB as the building and H as the helicopter hovering over it P is a point on the ground Angle of elevation of the top of building and helicopter are 300 and 600 We know that Height...
(i) In the adjoining figure, the angle of elevation from a point P of the top of a tower QR, 50 m high is 600 and that of the tower PT from a point Q is 300. Find the height of the tower PT, correct to the nearest metre. (ii) From a boat 300 metres away from a vertical cliff, the angles of elevation of the top and the foot of a vertical concrete pillar at the edge of the cliff are 550 40’ and 540 20’ respectively. Find the height of the pillar correct to the nearest metre.
Solution: Consider CB as the cliff and AC as the pillar D as the boat which is 300 m away from the foot of the cliff BD = 300 m Angle of elevation of the top and foot of the pillar are 550 40’ and...
An observer 1.5 m tall is 20.5 metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from the eye of the observer.
Solution: In the figure, AB is the tower and CD is an observer θ is the angle of observation It is given that AB = 22m CD = 1.5 m Distance BD = 20.5 m From the point C construct CE parallel tp DB AE...
The upper part of a tree broken by wind falls to the ground without being detached. The top of the broken part touches the ground at an angle of 380 30’ at a point 6 m from the foot of the tree. Calculate (i) the height at which the tree is broken. (ii) the original height of the tree correct to two decimal places.
Consider TR as the total height of the tree TP as the broken part which touches the ground at a distance of 6 m from the foot of the tree which makes an angle of 380 30’ with the ground Take PR = x...
A vertical tower is 20 m high. A man standing at some distance from the tower knows that the cosine of the angle of elevation of the top of the tower is 0.53. How far is he standing from the foot of the tower?
Solution: Consider AB as the tower Take a man C stands at a distance x m from the foot of the tower cos θ = 0.53 We know that Height of the tower AB = 20 m cos θ = 0.53 So we get θ = 580 Let us take...
A bridge across a river makes an angle of 450 with the river bank. If the length of the bridge across the river is 200 metres, what is the breadth of the river. olution: Consider AB as the width of...
An electric pole is 10 m high. A steel wire tied to the top of the pole is affixed at a point on the ground to keep the pole upright. If the wire makes an angle of 450 with the horizontal through the foot of the pole, find the length of the wire.
Consider AB as the pole and AC as the wire which makes an angle of 450 with the ground. Height of the pole AB = 10 m Consider x m as the length of wire AC We know that sin θ = AB/AC Substituting the...
A boy is flying a kite with a string of length 100 m
Consider AB as the height of the kite A and AC as the string Angle of elevation of the kite = 260 32’ Take AB = x m and AC = 100 m We know that sin θ = AB/AC Substituting the values sin 260 32’ =...
From the top of a cliff 92 m high, the angle of depression of a buoy is 200. Calculate to the nearest metre, the distance of the buoy from the foot of the cliff.
Solution: Consider AB as the cliff whose height is 92 m So we get x = 92 × 2.7475 x = 252.7700 m Hence, the distance of the buoy from the foot of the cliff is 252.77 m.
From a point P on level ground, the angle of elevation of the top of a tower is 300. If the tower is 100 m high, how far is P from the foot of the tower?
Solution: Consider AB as the tower and P is at a distance of x m from B which is the foot of the tower. Height of the tower = 100 m Hence, the distance of P from the foot of the tower is 173.2 m....
A river is 60 m wide. A tree of unknown height is on one bank. The angle of elevation of the top of the tree from the point exactly opposite to the foot of the tree on the other bank is 300. Find the height of the tree
Solution: Consider AB as the tree and BC as the width of the river Hence Proved
What is the angle of elevation of the sun when the length of the shadow of a vertical pole is equal to its height.
Solution: Consider AB as the pole and CB as its shadow θ is the angle of elevation of the sun Take AB = x m and BC = x m We know that tan θ = AB/CB = x/x = 1 So we get Hence...
A ladder is placed against a wall such that it reaches the top of the wall. The foot of the ladder is 1.5 metres away from the wall and the ladder is inclined at an angle of 600 with the ground. Find the height of the wall
Solution: Consider AB as the wall and AC as the ladder whose foot C is 1.5 m away from B Hence, the height of wall is 2.6 m.
The angle of elevation of the top of a tower from a point on the ground and at a distance of 150 m from its foot is 300. Find the height of the tower correct to one place of decimal.
Consider BC as the tower and A as the point on the ground such that So we get Hence Proved
An electric pole is 10 metres high. If its shadow is 10√3 metres in length, find the elevation of the sun.
Consider AB as the pole and OB as its shadow. It is given that Hence Proved