knowledge-route Maths10 Cha3


title: “Lata knowledge-route-Class10-Math1-2 Merged.Pdf(1)” type: “reveal” weight: 1

HEIGHTS & DISTANCES

HEIGHTS & DISTANCES

12.1 ANGLE OF ELEVATION :

In order to see an object which is at a higher level compared to the ground level we are to look up. The line joining the object and the eye of the observer is known as the line sight and the angle which this line of sight makes with the horizontal drawn through the eye of the observer is known as the angle of elevation. Therefore, the angle of elevation of an object helps in finding out its height (figure)

HEIGHTS & DISTANCES

12.2 ANGLE OF DEPRESSION :

When the object is at a lower level tan the observer’s eyes, he has to look downwards to have a view of the object. It that case, the angle which the line of sight makes with the horizontal thought the observer’s eye is known as the angle of depression (Figure).

HEIGHTS & DISTANCES

HEIGHTS & DISTANCES

ILLUSTRATIONS :

Ex. 1 A man is standing on the deck of a ship, which is 8m above water level. He observes the angle of elevations of the top of a hill as 60 and the angle of depression of the base of the hill as 30. Calculation the distance of the hill from the ship and the height of the hill.

[CBSE - 2005]

HEIGHTS & DISTANCES

Sol. Let x be distance of hill from man and h+8 be height of hill which is required. is right triangle ACB.

tan60=ACBC=hx3=hx

HEIGHTS & DISTANCES

In right triangle BCD.

13=8xx=83 Height of hill =h+8=3x+8=(3)(83)+8=32m. Distance of ship from hill =x=83m.

alt text

HEIGHTS & DISTANCES

Ex. 2 A vertical tower stands on a horizontal plane and is surmounted by vertical flag staff of height 5 meters. At a point on the plane, the angle of elevation of the bottom and the top of the flag staff are respectively 30 and 60 find the height of tower.

[CBSE-2006]

HEIGHTS & DISTANCES

Sol. Let AB be the tower of height h metre and BC be the height of flag staff surmounted on the tower, Let the point of the place be D at a distance x meter from the foot of the tower in ABD

tan30=ABAD13=hxx=3h.(i)

HEIGHTS & DISTANCES

In ABDtan60=ACAD

3=5+hxx=5+h3 …………..(ii)

HEIGHTS & DISTANCES

From (i) and (ii)

3h5+h33h=5+h2h=5h=52=2.5m So, the height of tower =2.5m

alt text

HEIGHTS & DISTANCES

Ex. 3 The angles of depressions of the top and bottom of 8m tall building from the top of a multistoried building are 30 and 45 respectively. Find the height of multistoried building and the distance between the two buildings.

HEIGHTS & DISTANCES

Sol. Let AB be the multistoried building of height h and let the distance between two buildings be x meters.

XAC=ACB=45 [Alternate angles AX||DE]

XAD=ADE=30 [Alternate angles AX||BC]

In ADE

tan30=AEED

13=h8x

(CB=DE=x)

HEIGHTS & DISTANCES

x=3(h8) …………(i)

In ACB

tan45=hx

1=hxx=h …………(ii)

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Form (i) and (ii)

3(h8)=h3h83=h3hh=83h(31)=83h=8331×(3+1)3+1h=83(3+1)2h=43(3+1)h=4(3+3) metres 

Form (ii) x=h

So, x=4(3+3) metres Hence, height of multistoried building =4(3+3) metres

Distance between two building =4(3+3) metres

HEIGHTS & DISTANCES

Ex. 4 The angle of elevation of an aeroplane from a point on the ground is 45. After a flight of 15sec, the elevation changes to 30. If the aeroplane is flying at a height of 3000 metres, find the speed of the aeroplane.

HEIGHTS & DISTANCES

Sol. Let the point on the ground is E which is y metres from point B and let after 15sec flight it covers x metres distance.

In AEB.

tan45=ABEB1=3000yy=3000m ………..(i)

In CED

tan30=CDED

13=3000x+y

(AB=CD)

x+y=30003 …………(ii)

HEIGHTS & DISTANCES

From equation (i) and (ii)

x+3000=30003x=300033000x=3000(31)

alt text

x=3000×(1.7321)x=2196m

HEIGHTS & DISTANCES

Hence, the speed of aeroplane is 527.04Km/hr.

HEIGHTS & DISTANCES

Ex. 5 If the angle of elevation of cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the distance of the cloud from the point of observation is 2hsecαtanβtanα.

HEIGHTS & DISTANCES

Sol. Let AB be the surface of the lake and let C be a point of observation such that ACh metres. Let D be the position of the cloud and D be its reflection in the lake. Then BD=BD.

In DCE

tanα=DECECE=Htanα.(i)

In Δ CED

tanβ=EECCE=h+H+htanβCE=2h+Htanβ.(ii)

HEIGHTS & DISTANCES

From (i) & (ii)

Htanα=2h+HtanβHtanβ=2htanα+HtanαHtanβHtanα+2htanαH(tanβtanα)=2htanαH=2htanαtanβtanα.(iii) In DCE Sinα=DECDCD=DEsinαCD=Hsinα

HEIGHTS & DISTANCES

Substituting the value of H from (iii)

CD=2htanα(tanβtanα)sinαCD=2hsinαcosα(tanβtanα)sinαCD=2htanαtanβtanα

alt text

Hence, the distance of the cloud from the point of observation is 2hsecαtanβtanα

Hence Proved.

HEIGHTS & DISTANCES

Ex. 6 A boy is standing on the ground and flying a kite with 100m of string at an elevation of 30. Another boy is standing on the roof of a 10m high building and is flying his kite at an elevation of 45. Both the boys are on opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet.

HEIGHTS & DISTANCES

Sol. Let the length of second string b×m.

In ABC

sin30=ACAB

12=AC100AC=50m

HEIGHTS & DISTANCES

In AEF

Sin30=AFAE

12=ACFCx

12=5010x[AC=50m,FC=ED=10m]12=40x

x=402m (So the length of string that the second boy must have so that the two kites meet =402m.)

DAILY PRACTICE PROBLEMS 12

OBJECTIVE DPP - 12.1

1. Upper part of a vertical tree which is broken over by the winds just touches the ground and makes an angle of 30 with the ground. If the length of the broken part is 20 metres, then the remaining part of the trees is of length

(A) 20 metres

(B) 103 metres

(C) 10 metres

(D) 102 metres

HEIGHTS & DISTANCES

Que. 1
Ans. C

HEIGHTS & DISTANCES

2. The angle of elevation of the top of a tower as observed from a point on the horizontal ground is ’ x ‘. If we move a distance ’ d ’ towards the foot of the tower, the angle of elevation increases to ’ y ‘, then the height of the tower is

(A) dtanxtanytanytanx

(B) d(tany+tanx)

(C) d(tanytanx)

(D) dtanxtanytany+tanx

HEIGHTS & DISTANCES

Que. 2
Ans. A

HEIGHTS & DISTANCES

3. The angle of elevation of the top of a tower, as seen from two points A & B situated in he same line and at distances ’ p ’ and ’ q ’ respectively from the foot of the tower, are complementary, then the height of the tower is

(A) pq

(B) pq

(C) pq

(D) noen of these

HEIGHTS & DISTANCES

Que. 3
Ans. C

HEIGHTS & DISTANCES

4. The angle of elevation of the top of a tower at a distance of 5033 metres from the foot is 60. Find the height of the tower

(A) 503 metres

(B) 203 metres

(C) -50 metres

(D) 50 metres

HEIGHTS & DISTANCES

Que. 4
Ans. D

HEIGHTS & DISTANCES

5. The Shadow of a tower, when the angle of elevation of the sun is 30, is found to be 5m longer than when its was 45, then the height of tower in metre is

(A) 53+1

(B) 52(31)

(C) 52(3+1)

(D) None of these.

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Que. 5
Ans. C

HEIGHTS & DISTANCES

SUBJECTIVE DPP - 12.2

1. From the top a light house, the angles of depression of two ships of the opposite sides of it are observed to be α and β. If the height of the light house be h meters and the line joining the ships passes thought the foot of the light house. Show that the distance between the ships is h(tanα+tanβ)tanαtanβ meters.

HEIGHTS & DISTANCES

2. A ladder rests against a wall at angle α to the horizontal. Its foot is pulled away from the previous point through a distance ’ a ‘, so that is slides down a distance ’ b ’ on the wall making an angle β. With the horizontal show that ab=cosαcosβsinβsinα

HEIGHTS & DISTANCES

3. From an aeroplanne vertically above a straight horizontal road, the angle of depression of two consecutive kilometer stone on opposite side of aeroplane are observed to be α and β. Show that the height of aeroplane above the road is tanαtanβtanα+tanβ kilometer.

HEIGHTS & DISTANCES

4. A round balloon of radius ’ r ’ subtends an angle θ at the eye of an observer while the angle of elevation of its centre is ϕ. Prove that the height of the centre of the balloon is rsinϕ cosec θ2.

HEIGHTS & DISTANCES

5. A window in a building is at a height of 10m from the ground. The angle of depression of a point P on the ground from the window is 30. The angle of elevation of the top of the building from the point P is 60. Find the height of the building.

HEIGHTS & DISTANCES

Sol. 5. 30 m

HEIGHTS & DISTANCES

6. A man on a cliff observers a boat at an angle of depression of 30 which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression

HEIGHTS & DISTANCES

Sol. 6. 9 min

HEIGHTS & DISTANCES

7. The angles of elevation of the top of a tower two points ’ P ’ and ’ Q ’ at distances of ’ a ’ and ’ b ’ respectively of the boat is found to be 60. Find the total time taken by the boat from the initial point to reach the shore.

from the base and in the same straight line with it, are complementary. Prove that the height of the tower is ab.

[CBSE - 2004]

HEIGHTS & DISTANCES

8. Two pillars of equal height are on either side of a road, which is 100m wide. The angles of elevation of the top the pillars are 60 and 30 at a point on the road between the pillar. Find the position of the pint between the pillars. Also find the height of each pillar,

[CBSE - 2005]

HEIGHTS & DISTANCES

Sol. 8. Height =43.3 m, Position - point is 25 m  from 1st  end and 75 m from 2nd  end.

HEIGHTS & DISTANCES

9. At a point, the angle of elevation of a tower is such that its tangent is 512, On walking 240 mnearer the tower, the tangent to the angle of elevation becomes 34, Find the height of the tower.

[CBSE - 2006]

HEIGHTS & DISTANCES

Sol. 9. 225 m

HEIGHTS & DISTANCES

10. From a window ’ x ‘mtres high above the ground in a street, the angles of elevation and depression of the top and foot of the other hose on the opposite side of the street are α and β respectively, Show that the opposite house is x(1+tanαcotβ) metres.

[CBSE - 2006]

HEIGHTS & DISTANCES

11. A pole 5m high is fixed on the top of a towel, the angle of elevation of the top of the pole observed from a point ’ A ’ on the ground is 60 an the angle of depression the point ; A; from the top of the tower is 45 Find the height of the tower.

[CBSE - 2007]

HEIGHTS & DISTANCES

Sol. 11. 6.82 m

HEIGHTS & DISTANCES

12. The angle of elevation of a jet fighter from a point A on the ground is 60 After a flight of 15 seconds, the angle O elevation changes to 30 If the jet is flying at a spies of 720km/fr, find the constant height at which the jet is flying. [use 3=1.732 ]

[CBSE - 2008]

HEIGHTS & DISTANCES

Sol. 12. 2598 m



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