Question

Two poles of equal heights are standing opposite each other on either side of the road which 80 ft width.From a point between them on the road the angle of elevation of the top of the poles are 60° and 30°.Find the height of the pole and the distance of the point from the pole

Answer 1

Given:-

AB and CD be the two poles of equal height.

Their heights be H m.

BC be the 80 m wide road.

P be any point on the road.

Let ,

CP be x m,

BP = (80 – x) . 

Also, ∠APB = 60° and ∠DPC = 30°

In right angled triangle DCP, 

Tan 30° = CD/CP 

⇒ h/x = 1/√3 

⇒ h = x/√3 ---------- (1) 

In right angled triangle ABP

Tan 60° = AB/AP 

⇒ h/(80 – x) = √3

⇒ h = √3(80 – x) 

⇒ x/√3 = √3(80 – x) 

⇒ x = 3(80 – x) 

⇒ x = 240 – 3x

⇒ x + 3x = 240

⇒ 4x = 240

⇒ x = 60 

Height of the pole, h = x/√3 = 60/√3 = 20√3. 

Thus, position of the point P is 60 m from C and height of each pole is 20√3 m.

hope it helps you

Answer 2

$\huge\mathfrak{Bonjour!!}$

$\huge\fcolorbox{black}{aqua}{Solution:-}$

✒ ʟᴇᴛ ᴀʙ ᴀɴᴅ ᴄᴅ ʙᴇ ᴛᴡᴏ ᴘᴏʟᴇs ᴏғ ᴇϙᴜᴀʟ ʜᴇɪɢʜᴛs ʜ ᴍᴇᴛʀᴇ ᴀɴᴅ ʟᴇᴛ ᴘ ʙᴇ ᴀɴʏ ᴘᴏɪɴᴛ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴘᴏʟᴇs, sᴜᴄʜ ᴛʜᴀᴛ

ᴀɴɢʟᴇ ᴀᴘʙ = 60° ᴀɴᴅ ᴀɴɢʟᴇ ᴅᴘᴄ = 30°.

ᴛʜᴇ ᴅɪsᴛᴀɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴛᴡᴏ ᴘᴏʟᴇs ɪs 80 ᴍ. (ɢɪᴠᴇɴ)

ʟᴇᴛ ᴀᴘ = x ᴍ, ᴛʜᴇɴ ᴘᴄ = (80 - x) ᴍ.

ɴᴏᴡ, ɪɴ ∆ᴀᴘʙ, ᴡᴇ ʜᴀᴠᴇ

ᴛᴀɴ 60° = ᴀʙ/ᴀᴘ = ʜ/x

=> √3 = ʜ/x

=> ʜ = √3 x .....→ (ɪ)

ᴀɢᴀɪɴ ɪɴ ∆ᴄᴘᴅ, ᴡᴇ ʜᴀᴠᴇ

ᴛᴀɴ 30° = ᴅᴄ/ᴘᴄ = ʜ/(80 - x)

=> 1/√3 = ʜ/80 - x

=> ʜ = 80 - x /√3 .....→ (ɪɪ)

ғʀᴏᴍ (ɪ) ᴀɴᴅ (ɪɪ), ᴡᴇ ʜᴀᴠᴇ

√3x = 80 - x/√3

=> 3x = 80 - x

=> 4x = 80

=> x = 80/4 = 20 ᴍ.

ɴᴏᴡ, ᴘᴜᴛᴛɪɴɢ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ x ɪɴ ᴇϙᴜᴀᴛɪᴏɴ (ɪ), ᴡᴇ ʜᴀᴠᴇ,

ʜ= √3 × 20 = 20√3

ʜᴇɴᴄᴇ, ᴛʜᴇ ʜᴇɪɢʜᴛ ᴏғ ᴛʜᴇ ᴘᴏʟᴇ ɪs 20√3ᴍ ᴀɴᴅ

ᴛʜᴇ ᴅɪsᴛᴀɴᴄᴇ ᴏғ ᴛʜᴇ ᴘᴏɪɴᴛ ғʀᴏᴍ ғɪʀsᴛ ᴘᴏʟᴇ ɪs

20 ᴍ ᴀɴᴅ ᴛʜᴀᴛ ᴏғ ᴛʜᴇ sᴇᴄᴏɴᴅ ᴏɴᴇ ɪs 60 ᴍ.

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