RD SHARMA Solutions for Class 10 Maths Chapter 12 - Heights and Distances

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Chapter 12 - Some Applications of Trigonometry Exercise Ex. 12.1

Solution 1

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Solution 5

Solution 6

Solution 7

Solution 8


Solution 9
Solution 10
Solution 11
Solution 12
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Solution 22
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Solution 24


Let BC be the building, AB be the transmission tower, and D be the point on ground from where elevation angles are to be measured.

Solution 25

Solution 26


Let AB be the statue, BC be the pedestal and D be the point on ground from where elevation angles are to be measured.

Solution 27


Solution 28
Solution 29


Let AB be the lighthouse and the two ships be at point C and D respectively.

Solution 30
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Solution 40

 

 

  

 

 

Solution 41

 

 

Let AC = h be the height of the chimney.

Height of the tower = DE = BC = 40 m

 

In ABE,

 

AB = BE√3….(i)

 

In ∆CBE,

tan 30° =

 

Substituting BE in (i),

AB = 40√3 × 3

= 120 m

 

Height of the chimney = AB + BC = 120 + 40 = 160 m

 

Yes, the height of the chimney meets the pollution control norms.

Solution 42

Let the ships be at B and C.

 

In D ABD,

 

BD = 200 m

 

In D ADC,

 

 

Distance between the two ships = BC = BD + DC

 

Solution 43

Here mCAB = mFEB = 30°.

Let BC = h m, AC = x m

In D ADE,

 

 

In D BAC,

 

 

Height of the second pole is 15.34 m

Solution 44



Solution 45



Let AQ be the tower and R, S respectively be the points which are 4m, 9m away from base of tower.


As the height can not be negative, the height of the tower is 6 m.

Solution 46

Solution 47



Solution 48

Solution 49



Solution 50



Solution 51




Solution 52



Solution 53




Solution 54




Solution 55



Solution 56 (i)

  

 

Let AB be the cliff, so AB=150m.

C and D are positions of the boat.

DC is the distance covered in 2 min.

ACB = 60o and ADB = 45o

ABC = 90o

In ΔABC,

tan(ACB)=

In ΔABD,

tan(ADB)=

So, DC=BD - BC

  =

Now,

begin mathsize 12px style speed equals distance over time
equals fraction numerator begin display style fraction numerator 50 open parentheses 3 minus square root of 3 close parentheses over denominator 1000 end fraction end style km over denominator 2 cross times begin display style 1 over 60 end style hrs end fraction... take space square root of 3 equals 1.732
equals 1.9019 space km divided by hr
equals 1902 space straight m divided by hr end style

Solution 56 (ii)

Let AB be the lighthouse and C be the position of man initially.

Suppose, a man changes his position from C to D.

As per the question, we obtain the following figure

Let speed of the boat be x metres per minute.

Therefore, CD = 2x

Using trigonometry, we have

Also,

Hence, speed of the boat is 57.8 m.

Solution 57

  

 

AB is the tower.

DC is the distance between cars.

AB=120m

In ΔABC,

tan(ACB) =

In ΔABD,

tan(ADB) =

So, DC=BD+BC

Solution 58

  

Let CD be the tower.

So CD =15m

AB is the distance between the points.

CAD = 60o and CBD = 45o

ADC = 90o

In ΔADC,

tan(CAD)=

In ΔCBD,

tan(CBD)=

So AB=BD - AD

Solution 59




Now, in triangle APB,

sin 60o = AB/ BP

√3/2 = h/ BP

This gives

h = 14.64 km

Solution 60



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Solution 66

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Solution 77

 

Solution 78

 

Chapter 12 - Some Applications of Trigonometry Exercise 12.41

Solution 1

begin mathsize 12px style straight theta space is space angle space of space elevation
given space AB over BC equals space fraction numerator 1 over denominator square root of 3 end fraction space space space space space space space..... open parentheses 1 close parentheses
We space know space tan space straight theta space equals space AB over BC space space space space space..... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
tan space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction space space also space tan 30 degree space equals space fraction numerator 1 over denominator square root of 3 end fraction
Hence space straight theta space equals space 30 degree
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 2

begin mathsize 12px style Let space straight h space be space the space height space of space tower
tan space 60 degree space equals space straight h over BC
straight h space equals space BC space tan space 60 degree
space space space space equals space 100 square root of 3 space straight m
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 3

begin mathsize 12px style AB space equals space Shadow space length
AB space equals space 30 space straight m
straight theta space equals space angle space of space elevation
straight theta space equals space 60 degree
tan space straight theta space equals space AC over AB
tan space 60 degree space equals space AC over 30
AC space equals space 30 square root of 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses tan 60 degree space equals space square root of 3 close parentheses
height space of space tower space equals space 30 square root of 3
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 4

begin mathsize 12px style BD space equals space straight a
BC space equals space straight b
tan space 60 degree space equals space AB over BC space and space tan space 30 degree space equals space AB over BD
square root of 3 equals AB over straight b space space space space space space space space..... open parentheses 1 close parentheses space and space fraction numerator 1 over denominator square root of 3 end fraction equals AB over straight a space space space space space.... open parentheses 2 close parentheses
open parentheses 1 close parentheses space cross times space open parentheses 2 close parentheses
square root of 3 cross times fraction numerator 1 over denominator square root of 3 end fraction equals AB squared over ab
AB space equals space square root of ab
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 5

begin mathsize 12px style Angles space are space complementery. space Hence space if space one space is space straight theta space other space must space be space 90 minus space straight theta
AB space equals space straight a
AC space equals space straight b
Let space the space height space AD space equals space straight h
In space triangle ABD
tan space straight theta space equals space AD over AB space equals space straight h over straight a space space space space space space space.... open parentheses 1 close parentheses
In space triangle ACD
tan space open parentheses 90 space minus space straight theta close parentheses space equals space AD over AC space equals space straight h over straight b space space space space space.... open parentheses 2 close parentheses
We space know space tan space open parentheses 90 space minus space straight theta close parentheses space equals space cot space straight theta space and space cot space straight theta space equals space fraction numerator 1 over denominator tan space straight theta end fraction
so space tan open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 1 over denominator tan space straight theta end fraction space space space space space space space space space space space space..... open parentheses 3 close parentheses
from space open parentheses 2 close parentheses space & space open parentheses 3 close parentheses
fraction numerator 1 over denominator tan space straight theta end fraction space equals space straight h over straight b
tan space straight theta space equals space straight b over straight h space space space space space space space space space space..... open parentheses 4 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 4 close parentheses
straight h over straight a equals space straight b over straight h
straight h squared space equals space ab
straight h space equals space square root of ab
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 6

begin mathsize 12px style MN space vertical line vertical line space DC
Hence space angle MAC space equals space angle ACB
and space angle NAD space equals space angle ADB
so space angle straight c space equals space 45 degree space and space angle straight D space equals space 30 degree
tan space straight C space equals space AB over BC space space space space space space space space space space space space space space space space space space space space space space space space space space space tan space straight D space equals space AB over BD
tan space 45 degree space equals space straight h over BC space space space space space space space space space space space space space space space space space space tan space 30 degree space equals space straight h over BD
BC space equals space straight h space space space space space space space space..... open parentheses 1 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space BD space equals space square root of 3 straight h space space space space space space space space....... open parentheses 2 close parentheses
distance space between space ships space equals space DB space plus space BC
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight h space plus space square root of 3 straight h
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses 1 space plus space square root of 3 close parentheses straight h
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 7

begin mathsize 12px style Let space CD space equals space straight h
tanα space equals space CD over CA space equals space fraction numerator straight h over denominator CB space plus space BA end fraction
tan space straight alpha space equals space fraction numerator straight h over denominator CB space plus space straight d end fraction
CB space plus space straight d space equals space fraction numerator straight h over denominator tan space straight alpha end fraction space space space space space space space space space space space space space space space..... open parentheses 1 close parentheses
tan space straight beta space equals space straight h over CB
CB space equals space fraction numerator straight h over denominator tan space straight beta end fraction space space space space space space space space space space space space.... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
fraction numerator straight h over denominator tan space straight beta end fraction plus space straight alpha space equals fraction numerator straight h over denominator tan space straight alpha end fraction
straight alpha space equals space hcot space straight alpha space minus space hcot space straight beta
space space space space space equals space straight h open parentheses cot space straight alpha space minus space cot space straight beta close parentheses
straight h equals space fraction numerator straight d over denominator cot space straight alpha space minus space cot space straight beta end fraction
So comma space the space correct space option space is space left parenthesis straight b right parenthesis.
end style

Solution 8

Wire BD

ED || AC

So, EA = DC and ED = AC

EA = 14

AB = EA + EB

20 = 14 + EB

EB = 6

begin mathsize 12px style In space triangle EDB
sin space 30 degree space equals space EB over BD
sin space 30 degree space equals space fraction numerator EB over denominator length space of space wire end fraction
length space of space wire space equals space fraction numerator EB over denominator sin space 30 degree end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 6 over denominator open parentheses 1 divided by 2 close parentheses end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 12 space straight m end style

 So, the correct option is (a).

Solution 9

begin mathsize 12px style Let space height space of space tower space is space straight h
straight theta space rightwards arrow space angle space of space elevation space of space tower space top
straight alpha space rightwards arrow space angle space of space depression space of space tower space foot
given space straight theta space equals space straight alpha
EC space vertical line vertical line space AB space and space EC space equals space AB space space space space space space space space space space space space space space space space space...... open parentheses 1 close parentheses
so space angle ECA space equals space angle CAB space space open parentheses Alternate space Interior space angle close parentheses
also space AD space parallel to space BC
Hence space AE space equals space BC space equals space 25 space straight m space space space space space space space space space..... open parentheses 2 close parentheses
Now space in space triangle ABC
tan space straight alpha space equals space BC over AB
tan space straight alpha space equals space 25 over AB space space space space space space space space space space space space.... open parentheses 3 close parentheses
Now space in space triangle space DEC
tan space straight theta space equals space ED over EC
tan space straight alpha space equals space fraction numerator AD space minus space AE over denominator EC end fraction
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
tan space straight alpha space equals space fraction numerator straight h space minus space 25 over denominator AB end fraction space and space tan space straight alpha space equals space 25 over AB
AB space tan space straight alpha space equals space straight h space minus space 25 space space space space space space space space space space space.... open parentheses 4 close parentheses
and space AB space tan space straight alpha space equals space 25 space space space space space space space space space space space space.. open parentheses 5 close parentheses
from space open parentheses 4 close parentheses space & space open parentheses 5 close parentheses
straight h space minus space 25 space equals space 25
straight h space equals space 50 space straight m
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 10

begin mathsize 12px style By space Alternate space interior space angle space property
angle straight B space equals space 45 degree space and space angle straight C space equals space 30 degree
Let space the space height space be space straight h
Now space tan space 30 degree space equals space straight h over AC space
AC space equals space square root of 3 straight h
AB space plus space BC space equals space square root of 3 straight h space space space space space space space space space space space space space space space space..... open parentheses 1 close parentheses
tan space 45 degree space equals space straight h over AB
AB space equals space straight h
AB space equals space straight h space space space space space space space space space space space space space....... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space and space open parentheses 2 close parentheses
straight h space plus space BC space equals space square root of 3 straight h
straight h space plus space 100 space equals space square root of 3 straight h
100 space equals space open parentheses square root of 3 space minus space 1 close parentheses straight h
straight h space equals space fraction numerator 100 over denominator square root of 3 minus space 1 end fraction cross times fraction numerator square root of 3 space plus space 1 over denominator square root of 3 space plus space 1 end fraction equals fraction numerator 100 open parentheses square root of 3 space plus space 1 close parentheses over denominator 3 space minus space 1 end fraction
straight h space equals space 50 open parentheses square root of 3 space plus space 1 close parentheses
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Chapter 12 - Some Applications of Trigonometry Exercise 12.42

Solution 11

begin mathsize 12px style Let space cloud space is space at space height space straight h space from space lake comma space so space its space reflection space is space straight a space depth space straight h space from space the space lake.
OC space equals space OR space equals space straight h
AB space equals space OE space equals space 200 space straight m
In space triangle CEB
tan space 30 degree space equals space EC over EB
EB space equals space square root of 3 space EC space space space space space space space space space space space space space space space space space space space....... open parentheses 1 close parentheses
In space triangle BER
tan space 60 degree space equals space RE over EB
EB space equals space fraction numerator RE over denominator square root of 3 end fraction space space space space space space space space space space space space space space space space space space space space space space space...... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
square root of 3 EC space equals space fraction numerator RE over denominator square root of 3 end fraction
3 open parentheses OC space minus space OE close parentheses space equals space OR space plus space OE
3 open parentheses straight h space minus space 200 close parentheses space equals space straight h space plus space 200
3 straight h space minus space 600 space equals space straight h space plus space 200
2 straight h space equals space 800
straight h space equals space 400 space straight m
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style

Solution 12

begin mathsize 12px style CD space equals space straight x
tan space 30 degree space equals space 100 over BD space and space tan space 45 degree space equals space AB over BC equals space 100 over BC
BD space equals space 100 square root of 3 space space space space space space and space space BC space equals space 100
BC space plus space CD space equals space 100 square root of 3
100 space plus space straight x space equals space 100 square root of 3
straight x space equals space 100 open parentheses square root of 3 space minus space 1 close parentheses space straight m
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 13

If height of one person is x then height of another one is 2x. Also If angle of elevation of one is θ then for another it is 90 - θ.

AB = a

C is mid point.

begin mathsize 12px style So space AC space equals space BC space equals space straight a over 2
tan space straight theta space equals space fraction numerator straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals fraction numerator 2 straight x over denominator straight a end fraction space space space space space space space.... open parentheses 1 close parentheses
tan open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 2 straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals space fraction numerator 4 straight x over denominator straight a end fraction space space space space space space space space space space.... open parentheses 2 close parentheses
We space know comma space tan space open parentheses 90 space minus space straight theta close parentheses space equals space cotθ
space space space space space space space space space space space space space space space space space space space space space space tan space open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 1 over denominator tan space straight theta end fraction
space space space space space space space space space space space space space space space space space space space space space space tan space straight theta space. space tan space open parentheses 90 space minus space straight theta close parentheses space equals space 1 space space space space space space space space space space space...... open parentheses 3 close parentheses
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
open parentheses fraction numerator 2 straight x over denominator straight a end fraction close parentheses open parentheses fraction numerator 4 straight x over denominator straight a end fraction close parentheses space equals space 1
fraction numerator 8 straight x squared over denominator straight a squared end fraction equals 1
straight x squared space equals space straight a squared over 8 space space space space space space space space space rightwards double arrow straight x equals fraction numerator straight a over denominator 2 square root of 2 end fraction end style

So, the correct option is (d).

Solution 14

begin mathsize 12px style Correct space option space colon space open parentheses straight a close parentheses
Let space height space of space cloud space from space lake space is space straight H. space So space reflection space is space at space straight H space depth space from space lake.
tan space straight theta space equals space DC over DA space space space space space space space space space space space space.... open parentheses 1 close parentheses
tan space 45 degree space equals space RD over DA space space space space space space space space space space..... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
fraction numerator DC over denominator tan space straight theta end fraction space equals space fraction numerator RD over denominator tan space 45 degree end fraction
fraction numerator OC space minus space OD over denominator tan space straight theta end fraction space equals space fraction numerator RO space plus space OD over denominator 1 end fraction
fraction numerator straight H space minus space straight h over denominator tan space straight theta end fraction space equals space straight H space plus space straight h
straight H space minus space straight h space equals space straight H space tan space straight theta space plus space straight h space tan space straight theta
straight H open parentheses 1 space minus space tan space straight theta close parentheses space equals space straight h open parentheses 1 space plus space tan space straight theta close parentheses
straight H space equals space fraction numerator straight h open parentheses 1 space plus space tan space straight theta close parentheses over denominator open parentheses 1 space minus space tan space straight theta close parentheses end fraction
straight H space equals space fraction numerator straight h open parentheses tan space 45 degree space plus space tan space straight theta close parentheses over denominator open parentheses 1 space minus space tan space straight theta space tan space 45 degree close parentheses end fraction
We space know space tan space open parentheses straight A space plus space straight B close parentheses space equals space fraction numerator tan space straight A space plus space tan space straight B over denominator 1 space minus space tan space straight A space tan space straight B end fraction
so space straight H space equals space straight h space tan space open parentheses 45 degree space plus space straight theta close parentheses
So comma space the space correct space option space is space left parenthesis straight a right parenthesis.
end style

Solution 15

begin mathsize 12px style Let space height space of space tower space be space straight H.
Angle space of space depression space is space 60 degree. space So space by space alternate space interior space angle space property.
angle DBC space equals space 60 degree
tan space 60 degree space equals space DC over BC
BC space equals space fraction numerator DC over denominator tan space 60 degree end fraction
space space space space space space space space equals space fraction numerator straight h over denominator square root of 3 end fraction space space space space space space space space.... open parentheses 1 close parentheses
tan space 30 degree space equals space AB over BC
BC space equals space fraction numerator AB over denominator tan space 30 degree end fraction
BC space equals space square root of 3 AB
BC space equals space square root of 3 straight H space space space space space space space space space space space space space space.... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
square root of 3 straight H space equals space fraction numerator straight h over denominator square root of 3 end fraction
straight H space equals space straight h over 3
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 16

begin mathsize 12px style Let space the space height space be space straight h.
tan space 30 degree space equals space straight h over AC space equals space fraction numerator straight h over denominator AB space plus space straight x end fraction
AB space plus space straight x space equals space straight h square root of 3 space space space space space space space space space.... open parentheses 1 close parentheses
tan space 60 degree space equals space straight h over AB
AB space equals space fraction numerator straight h over denominator square root of 3 end fraction space space space space space space space space space space space space..... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
fraction numerator straight h over denominator square root of 3 end fraction space plus space straight x space equals space straight h square root of 3
straight x space equals space straight h square root of 3 space minus space fraction numerator straight h over denominator square root of 3 end fraction
space space space space space space equals space fraction numerator 2 straight h over denominator square root of 3 end fraction
straight h space equals space fraction numerator square root of 3 over denominator 2 end fraction straight x
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 17

begin mathsize 12px style Let space height space be space straight h space and space BC space equals space 2 straight x
tan space 30 degree space equals space AD over AC space equals space fraction numerator straight h over denominator AB space plus space BC end fraction
AB space plus space 2 straight x space equals space straight h square root of 3 space space space space space space space space space space...... open parentheses 1 close parentheses
tan space 45 degree space equals straight h over AB
AB space equals space straight h space space space space space space space...... open parentheses 2 close parentheses
from space open parentheses 1 close parentheses space & space open parentheses 2 close parentheses
straight h space plus space 2 straight x space equals space straight h square root of 3
2 straight x space equals space straight h open parentheses square root of 3 space minus space 1 close parentheses
straight h space equals space fraction numerator 2 straight x over denominator square root of 3 space minus space 1 end fraction cross times space fraction numerator square root of 3 space plus space 1 over denominator square root of 3 space plus space 1 end fraction
space space space space space space equals space fraction numerator 2 open parentheses square root of 3 space plus space 1 close parentheses straight x over denominator 3 space minus space 1 end fraction
straight h space equals space open parentheses square root of 3 space plus space 1 close parentheses straight x
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 18

If height of one pole is x then height of the other one is 2x. Also If the angle of elevation of one is θ then for the other it is

90 - θ.

AB = a

C is mid point.

begin mathsize 12px style So space AC space equals space BC space equals space straight a over 2
tan space straight theta space equals space fraction numerator straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals fraction numerator 2 straight x over denominator straight a end fraction space space space space space space space.... open parentheses 1 close parentheses
tan open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 2 straight x over denominator open parentheses begin display style straight a over 2 end style close parentheses end fraction equals space fraction numerator 4 straight x over denominator straight a end fraction space space space space space space space space space space.... open parentheses 2 close parentheses
We space know comma space tan space open parentheses 90 space minus space straight theta close parentheses space equals space cotθ
space space space space space space space space space space space space space space space space space space space space space space tan space open parentheses 90 space minus space straight theta close parentheses space equals space fraction numerator 1 over denominator tan space straight theta end fraction
space space space space space space space space space space space space space space space space space space space space space space tan space straight theta space. space tan space open parentheses 90 space minus space straight theta close parentheses space equals space 1 space space space space space space space space space space space...... open parentheses 3 close parentheses
from space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses comma space open parentheses 3 close parentheses
open parentheses fraction numerator 2 straight x over denominator straight a end fraction close parentheses open parentheses fraction numerator 4 straight x over denominator straight a end fraction close parentheses space equals space 1
fraction numerator 8 straight x squared over denominator straight a squared end fraction equals 1
straight x squared space equals space straight a squared over 8 space space space space space space space space space rightwards double arrow straight x equals fraction numerator straight a over denominator 2 square root of 2 end fraction end style

So, the correct option is (b).

Solution 19

EC || AB

Hence

EA = CB = 10

AD = AE + ED

ED = AD - AE

     = 16 - 10 = 6

 begin mathsize 12px style In space triangle DEC
sin space 30 degree space equals space DE over straight l
straight l space equals space fraction numerator DE over denominator sin space 30 end fraction space rightwards double arrow space straight l space equals space 2 space cross times space 6 space equals space 12 end style

So, the correct option is (c).

Solution 20

begin mathsize 12px style AE space rightwards arrow space lamp space post
BD space rightwards arrow space girl
AB space equals space 3
BC space equals space 4.5 space straight m
tan space straight Q space equals space BD over BC space equals space AE over AC
space space space space space space space space space space space space space space space space space fraction numerator 1.5 over denominator 4.5 end fraction equals fraction numerator straight h over denominator AB space plus space BC end fraction
space space space space space space space space space space space space space space space space space space space 1 third equals fraction numerator straight h over denominator 3 space plus space 4.5 end fraction
space space space space space space space space space space space space space space space space space space space space space straight h space equals space fraction numerator 7.5 over denominator 3 end fraction equals 2.5
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 12 - Some Applications of Trigonometry Exercise 12.43

Solution 21

begin mathsize 12px style straight h space rightwards arrow space height space of space tower
straight s space rightwards arrow space shadow space of space tower
straight theta space rightwards arrow angle space of space elevation
Given colon space straight s space equals space square root of 3 straight h space rightwards double arrow straight h over straight s equals space fraction numerator 1 over denominator square root of 3 end fraction space space space space space space space space space space space..... open parentheses 1 close parentheses
In space the space given space triangle comma
tan space straight theta space equals space straight h over straight s
from space open parentheses 1 close parentheses
tan space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction
straight theta space equals space 30 degree
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 22

 

 

From the figure, it is cleared that we have to find the length of BC.

begin mathsize 12px style tan space straight theta space fraction numerator Opposite space side over denominator Adjacent space Side end fraction equals AB over BC
rightwards double arrow tan space 30 degree space equals space 75 over BC
rightwards double arrow fraction numerator 1 over denominator square root of 3 end fraction equals 75 over BC
rightwards double arrow BC space equals space 75 square root of 3
Correct space Option colon thin space straight C
end style

Solution 23

begin mathsize 12px style AB space rightwards arrow space wall
AC space rightwards arrow space ladder
angle straight A space plus space angle straight B space plus space angle straight C space equals space 180 degree
angle straight C space equals space 180 degree space minus space 90 degree space minus space 60 degree
space space space space space space space space equals space 30 degree
sin space 30 degree space equals space AB over AC
AB space equals space AC space sin space 30 degree
space space space space space space space equals space AC over 2
AB space equals space 15 over 2 straight m
So comma space the space correct space option space is space left parenthesis straight c right parenthesis. end style

Solution 24

begin mathsize 12px style angle straight C space equals space 30 degree
Hence comma
tan space 30 degree space equals space AB over BC rightwards double arrow square root of 3 space equals space AB over BC
rightwards double arrow BC space equals space fraction numerator 1 over denominator square root of 3 end fraction space AB
space space space space space space space space space space space equals space fraction numerator 1 over denominator square root of 3 end fraction space left parenthesis 150 right parenthesis equals 50 square root of 3
So comma space the space correct space option space is space left parenthesis straight a right parenthesis. end style

Solution 25

begin mathsize 12px style straight h space rightwards arrow space height space of space pole space
straight b space rightwards arrow space length space of space shadow
straight theta space rightwards arrow space angle space of space elevation
given space straight h space equals space square root of 3 space straight b
straight h over straight b space equals space square root of 3
tan space straight theta space equals straight h over straight b
tan space straight theta space equals space square root of 3
straight theta space equals space 60 degree
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 26

begin mathsize 12px style tan space 45 degree space equals space straight h over 50
rightwards double arrow 1 space equals space straight h over 50
rightwards double arrow straight h space equals space 50
So comma space the space correct space option space is space left parenthesis straight b right parenthesis. end style

Solution 27

begin mathsize 12px style Length space of space ladder space equals space BC
cos space 60 degree space equals space AC over BC equals 2 over BC
BC space equals space fraction numerator 2 over denominator cos space 60 degree end fraction space equals fraction numerator 2 over denominator begin display style 1 half end style end fraction equals space 4
So comma space the space correct space option space is space left parenthesis straight d right parenthesis. end style