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.
Answers
The height of the poles are 30√3 feet and 60√3 feet .
Step-by-step explanation:
Referring to the figure attached below, let's make some assumptions
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 feet
Substituting the value of BC in (i), we get
h1 = BC√3 = 30√3 feet
∴ h2 = 2 * h1 = 2 * 30√3 = 60√3 feet
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