Question

3. The angles of elevation of a lamppost
changes from 30° to 60° when a man walks
20 m towards it. What is the height of the
lamppost?
1) 8.66 m 2) 10 m 3) 17.32 m 4) 20 m​

Answer 1

Answer:

17.32

Step-by-step explanation:

tan30=1/sqrt(3)

y/(20+x)=1/sqrt(3)--------(1)

and also we have

tan60 =sqrt3

y/x=sqrt3-------(2)

solving (1) (2) we get y i.e., the height as 17.32

Answer 2

Question:

The angles of elevation of a lamppost changes from 30° to 60° when a man walks towards it. What is the height of the lamppost ?

1) 8.66 m

2) 10 m

3) 17.32 m

4) 20 m

Answer:

Option (3) => 17.32 m.

Note:

• In a right-angled triangle the side opposite to the right angle is its hypotenuse.
• The hypotenuse of a right-angled triangle is its longest side.
• In a right-angled triangle, the other two sides (other than the hypotenuse) are orthogonal.
• The word "orthogonal" means mutually perpendicular.
• In a right-angled triangle, the orthogonal side which lies opposite to an angle (other than the 90°) is considered to be the perpendicular for that angle and the another orthogonal side is considered to be its base.
• Pythagoras theorem: This theorem states that, in a right-angled triangle, the square of its hypotenuse is equal to the sum of squares of its other two orthogonal sides.

ie; h^2 = b^2 + p^2

• Angle of elevation: It is the angle formed between the horizontal line and the line of sight.
• The approx value of √3 is 1.732.
• tan@ = perpendicular/base
• tan@ = sin@/cos@
• tan0° = 0
• tan30° = 1/√3
• tan45° = 1
• tan60° = √3
• tan90° = ∞

Solution:

Let's plot a rough sketch to describe the situation given in the question.

Let a lamppost AB of height p m .

Initially the angle of elevation of the lamppost at the point D is 30°.

Let the man moves 20 m towards the lamppost from the point D to the point C forming the final angle of elevation of 60° at the point C.

Alos, let the distance between the foot of lamppost and the final position of the man be BC = x m.

{ For the plot ,refer to the attachment }

Now,

In the right-angled ∆ABD ,

=> tanD = AB/BD

=> AB = BD•tanD

=> AB = (BC + CD)•tanD

=> p = (x + 20)•tan30°

=> p = (x + 20)•(1/√3)

=> p = (x + 20)/√3 ----------(1)

Also,

In the right-angled ∆ABC ,

=> tanC = AB/BC

=> AB = BC•tanC

=> p = x•tan60°

=> p = x•(√3)

=> p = √3x -------------(2)

Now,

From eq-(1) and eq-(2) , we have;

=> (x + 20)/√3 = √3x

=> (x + 20) = √3•(√3x)

=> x + 20 = 3x

=> 3x - x = 20

=> 2x = 20

=> x = 20/2

=> x = 10

Now,

Putting x = 10 ,in eq-(2) , we get;

=> p = √3x

=> p = √3•10

=> p = 1.732•10

=> p = 17.32

Hence,

The height of the lamppost AB is ;

17.32 m.