Question

Two poles of equal heights are standing opposite each other on either sides 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.​

Answer 1

✬ Height of Poles = 20√3 m ✬

Distance of Point = 60 & 20 m ✬

Step-by-step explanation:

Given:

• Height of poles are equal.
• Width of road is 80 m.
• Angles of elevation of top of poles are 60° & 30°.

To Find:

• What is the height of poles and distances of point from the poles ?

Solution: Let the height of both poles (AB & EF) be h m.

• BF = road of 80 m.
• BF = BC + FC

Let BC be x m. Therefore, FC will be (80 – x) m.

Now in right angled ∆ABC , by using tanθ.

➟ tanθ = Perpendicular/base

➟ tan30° = AB/BC

➟ 1/√3 = h/x

➟ x = √3h.............i

Now in ∆EFC , by using tanθ

➟ tan60° = EF/FC

➟ √3 = h/(80 – x)

➟ √3(80 – x) = h

➟ 80√3 – √3x = h

➟ 80√3 – √3(√3h) = h

➟ 80√3 – 3h = h

➟ 80√3 = h + 3h

➟ 80√3 = 4h

➟ 80√3/4 = h

➟ 20√3 = h

Hence, the height of poles is 20√3 m.

Now put the value of h in equation i.

• x = √3 × 20√3 = 60 m

So, BC = x = 60 m and FC = 80 – 60 = 20 m.

Answer 2

Answer:

✯ Given :-

• Two poles of equal heights are standing opposite each other on either sides 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.

✯ To Find :-

• What is the height of the poles and the distance of the point from the poles.

✯ Solution :-

» Let, AD and BC be two poles of equal heights h m.

» And, P be a point on the road such that AP = x m, BP = (80 - x) m.

» Given that, ∠APD = 60°, ∠BPC = 30°

➙ In the ∆APD, we have,

⇒ tan60° = $\dfrac{AD}{AP}$

⇒ √3 = $\dfrac{h}{x}$

⇒ x = $\dfrac{h}{√3}$ ..... Equation no (1)

➙ In the ∆BPC we have,

⇒ tan30° = $\dfrac{BC}{BP}$

⇒ $\dfrac{1}{√3}$ = $\dfrac{h}{80 - x}$

⇒ 80 - x = √3h

⇒ x = 80 - √3h .... Equation no (2)

➣ By solving the equation no (1) and (2) we get,

⇒ $\dfrac{h}{√3}$ = 80 - √3 h

⇒ h = √3 (80 - √3h)

⇒ h = 80√3 - 3h

⇒ 4h = 80√3

⇒ h = $\sf\dfrac{\cancel{80√3}}{\cancel{4}}$

$\dashrightarrow$ h = 20√3

➣ Putting h = 20√3 in the equation no (1) we get,

⇒ x = $\dfrac{h}{√3}$

⇒ x = $\dfrac{20√3}{√3}$

➠ x = 20 m

⋆ And,

• AP = x = 20 m
• BP = 80 - x = 80 - 20 = 60 m

$\therefore$ The height of the poles is 20√3 m .

$\therefore$ The distance of the point from the poles is 20 m and 60 m .

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