Question

Question 1)

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles ..

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Answer 1

Height of Pole: 20√3 m

Distance Of Point From Poles: 20m and 60m

Explanation:

The attached graph may help you understand my explanation better

The two poles are Ab and CD and are 80m apart

O is the point between them

The angle of elevation from O to A is 60°

The angle of elevation from O to D is 30°

Since the two poles are given to be of equal height,

AB = CD

In ΔAOB using the law of tangents

tan 60° = AB/BO

tan 60° = √3 and AB = x

So we get

√3 = x/BO

Multiplying both sides by BO we get

√3 · BO = x

or BO = x/√3

Looking at ΔCOD

tan 30° = DC/OC

tan 30° = 1/√3

1/√3 = x/OC

OC · ( 1/√3) = x

OC = x ÷ 1/√3

= x · √3/1

OC = √3x

We also know that BO + OC = 80

Plugging this information with values computed gives us

x/√3 + √3x = 80

Multiply by √3 on both sides

(x/√3) · √3 + (√3x) √3 = 80√3

(x/√3) · √3 = x since the √3 terms cancel out

(√3x) √3 = 3x

So we get

(x/√3) · √3 + (√3x) √3 = 80√3

=>
x + 3x = 80√3

4x = 80√3

x = (80√3)/4

x =20√ 3

So the height of each pole = 20√ 3 m ≈ 34.64 m

The distance OC can be found from the previous equation
OC = √3x

=> OC = √3 x 20√3 = 20 x 3 = 60m

Since BO + OC = 80 m

BO = 80 - 60 = 20 m

Answer:
Height of each pole = 20√3 m

The point is located 20 m from one pole and 60 m from the other pole

Answer 2

The height of pole is 20√3 m each and distance of pole from the point on road is 60 m and 20 m respectively.

Explanation:

Given:

• Two poles of equal heights are standing on opposite side of the road which is 80 m wide facing each other.
• From a point on the road the angle of elevation of poles are 60° and 30° respectively.

To Find:

• The height of poles and the distance of that point on the road from the pole.

Formula used:

• tan60° = √3
• tan30° = 1/√3

Let the height of each pole be H metres.

Also, let's assume that distance of the point from one pole (as shown in figure) be x m.

According to question, we have

tan60° = H/(80-x)

H = (80-x)√3.... (1)

Also, tan30° = H/x

H = x/√3... (2)

Putting value of eq(2) in eq(1), we get

x/√3 = (80-x)√3

x = 3(80-x)

x = 240 - 3x

4x = 240

x = 60 m

Putting value of x in eq(2), we get

H = 60/√3 = 20√3 m

So, the height of pole is 20√3 m each and distance of pole from the point on road is 60 m and 20 m respectively.