Question

The height of two poles are in the ratio of 1:2 standing opposite to each other an either side of road which is 90 feet wide a man observes the top of the polesame from point on the line joining the foot of the poles if the angle of elevation is 60 find the length of poles

Answer 1

✪ Cᴏʀʀᴇᴄᴛ Qᴜᴇsᴛɪᴏɴ :-

A man observes two vertical poles which are fixed opposite to each other on either side of the road. If the width of the road is 90 feet and heights of the pole are in the ratio 1: 2, also the angle of elevation of their tops from a point between the line joining the foot of the poles on the road is 60°. Find the heights of the poles.

✪ Gɪᴠᴇɴ :-

• Width of the pole is 90 feet
• Height of the pol are in the ratio 1 : 2
• Angle of elevation of their tops from a point between the line joining the foot of the poles on the road is 60°

✪ Tᴏ Fɪɴᴅ :-

• Heights of the poles

✪ Sᴏʟᴜᴛɪᴏɴ :-

Refer the image first

• AB = h1 ( height of the 1st pole )
• ED = h2 ( height of the 2nd pole )
• BD = 90 feet ( distance between the 2 poles )

Angle of the elevation from point C to the top of AB = Ѳ1 = 60°

Angle of the elevation from point C to the top of ED = Ѳ2 = 60°

Let BC be the distance of the point C from the foot of AB, then CD = (90 - BC) be the distance of point C from the foot of ED. So, h1 is the height of the 1st pole AB, then h2 = 2h1 is the height of the 2nd pole ED

Now, In ΔABC

⟾ tan 60° = AB/BC

⟾ √3 = h1/BC

⟾ h1 = BC√3 _________①

Now, In ΔEDC

⟾ tan 60° = ED/CD

⟾ √3 = h2/(90-BC)

⟾ 2h1 = √3(90-BC)

⟾ h1 = (√3 / 2) (90 - BC)_________②

From eq ① & ②, we get

BC√3 = (√3 / 2) (90-BC)

⟾ 2BC = 90-BC

⟾ 2BC + BC = 90

⟾ 3BC = 90

⟾ BC = 30 Feet

Substituting BC in ①

h1 = BC√3

= 30√3 feet

h2 = 2 × h1

= 2 × 30√3

= 60√3 feet

⛬ The height of the poles are 30√3 feet and 60√3 feet.