A man observes two vertical poles which are fixed opposite on either side of the road. If the width of the road is 90 feet and heights of the poles 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.

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Step-by-step explanation:

AB = h₁= height of the first pole

ED = h₂ = height of the second pole

BD = distance between the two poles = 90 feet

Angle of elevation from point C to the top of AB = θ₁ = 60°

Angle of elevation from point C to the top of ED = θ₂ = 60°

Let the distance of point C from the foot of AB be “BC”, then the distance of point C from the foot of ED will be “CD = (90 - BC)”.

Since it is given that the ratio of the heights of the pole are 1:2

So, if the height of the first pole AB is “h1” then the height of the second pole ED will be "h2 = 2h1”.

Now,  

Consider ΔABC, applying the trigonometric ratios of a triangle, we get

tan θ₁ = perpendicular/base

⇒ tan 60° = AB/BC

⇒ √3 = h₁/BC

⇒ h₁ = BC√3 … (i)

and,

Consider ΔEDC, applying the trigonometric ratios of a triangle, we get

tan θ₂ = perpendicular/base

⇒ tan 60° = ED/CD

⇒ √3 = h₂/(90 - BC)

⇒ 2h₁ = √3 [90 - BC]

⇒ h₁ = (√3/2) [90 - BC] … (ii)

From (i) & (ii), we get

BC√3 = (√3/2) [90 - BC]

⇒ 2BC = 90 – BC

⇒ 2BC + BC = 90

⇒ 3BC = 90

⇒ BC = 90/3

⇒ BC = 30

Substituting the value of BC in (i), we get

h₁ = BC√3 = 30√3 feet  ← height of the first pole

∴ h₂ = 2 * h₁ = 2 * 30√3 = 60√3 feet  ← height of the second pole

Also View:

The angle of elevation of the cloud from a point 60m above the surface of the water of a lake is 30 degree and angle of depression of its Shadow from the same point in a water of lake is 60 degree find the height of the cloud from the surface of water ?

Two poles are 200 m apart if from the middle point of the line joining their feet an observer finds the angle of elevation of their tops as 60 and 30 degree respectively then find the ratio of the height of the poles.

Angles of elevation of the top of a tower from two points at distance of 9 m and 16 m from the base of the tower in the same side and in the same straight line with it are complementary. Find the height of the tower.

answer:

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