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Mathematics

Two poles of equal heights are standing opposite to each other on either side of a road, which is 30 m wide. From a point between them on the road, the angles of elevation of the tops are 30° and 60°. The height of each pole is:

  1. 4.33 m

  2. 6.5 m

  3. 13 m

  4. 15 m

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Answer

Two poles of equal heights are standing opposite to each other on either side of a road, which is 30 m wide. From a point between them on the road, the angles of elevation of the tops are 30° and 60°. The height of each pole is: Volume And Surface Area of solid RSA Mathematics Solutions ICSE Class 10.

Let AB and CD be two poles of equal height (h).

BD = 30 m

Let BP = x m

In triangle ABP,

⇒tan⁡60∘=ABBP⇒3=hx⇒x=h3\Rightarrow \tan 60^{\circ} = \dfrac{AB}{BP} \\[1em] \Rightarrow \sqrt3 = \dfrac{h}{x} \\[1em] \Rightarrow x = \dfrac{h}{\sqrt3}

In triangle PCD,

⇒tan⁡30∘=h30−x⇒13=h30−x⇒30−x=h3⇒30−h3=h3⇒303−h3=h3⇒303−h=h3(3)⇒303−h=3h⇒303=3h+h⇒303=4h⇒h=3034⇒h=7.53⇒h=7.5(1.732)⇒h=12.99≈13 m.\Rightarrow \tan 30^{\circ} = \dfrac{h}{30 - x} \\[1em] \Rightarrow \dfrac{1}{\sqrt3} = \dfrac{h}{30 - x} \\[1em] \Rightarrow 30 - x = h\sqrt3 \\[1em] \Rightarrow 30 - \dfrac{h}{\sqrt3} = h\sqrt3 \\[1em] \Rightarrow \dfrac{30\sqrt3 - h}{\sqrt3} = h\sqrt3 \\[1em] \Rightarrow 30\sqrt3 - h = h\sqrt3(\sqrt3) \\[1em] \Rightarrow 30\sqrt3 - h = 3h \\[1em] \Rightarrow 30\sqrt3 = 3h + h \\[1em] \Rightarrow 30\sqrt3 = 4h \\[1em] \Rightarrow h = \dfrac{30\sqrt3}{4} \\[1em] \Rightarrow h = 7.5\sqrt3 \\[1em] \Rightarrow h = 7.5(1.732) \\[1em] \Rightarrow h = 12.99 \approx 13 \text{ m.}

Hence, option 3 is the correct option.

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