Question

Two poles of equal heights are standing opposite each other on either side 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 distance of the point from the poles.
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Answer 1

$\large\underline{\underline{\rm{\red{Given:-}}}}$

• $\rm \angle APB = 60^0$
• $\rm \angle DPC = 30^0$
• $\rm BC = 80 \ m$

$\large\underline{\underline{\rm{\red{Solution:-}}}}$

Let BP = x , CP = 80 - x

Now , in $\rm \triangle ABP$

• $\rm :\longmapsto tan60^0 = \dfrac{AB}{BP}$

• $\rm :\longmapsto \sqrt3 = \dfrac{AB}{x}$

• $\rm :\longmapsto AB = \sqrt3x$

Now in $\rm \triangle BPC$

• $\rm :\longmapsto tan30^0 = \dfrac{CD}{CP}$

• $\rm :\longmapsto \dfrac{1}{\sqrt3} = \dfrac{CD}{80 - x}$

• $\rm :\longmapsto CD = \dfrac{80-x}{\sqrt3}$

As AB = CD,

• $\rm :\longmapsto \sqrt3 x = \dfrac{80-x}{\sqrt3}$
• $\rm :\longmapsto 3x = 80 - x$
• $\rm :\longmapsto 4x = 80$
• $\rm :\longmapsto x = 20\ m$

Hence, height of pole is given as:

$\boxed{\boxed{\rm h = \sqrt3x = 20\sqrt3\ m}}$

Distance of point from pole AB

• $\boxed{\boxed{\rm BP = x \ m = 20 \ m}}$

Distance of point from pole CD

• $\boxed{\boxed{\rm CP = 80 - x = 60 \ m}}$

More to know:

$\sf \color{aqua}{Trigonometry\: Table}\\ \blue{\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \sf \red{\angle A} & \red{\sf{0}^{ \circ} }&\red{ \sf{30}^{ \circ} }& \red{\sf{45}^{ \circ} }& \red{\sf{60}^{ \circ}} &\red{ \sf{90}^{ \circ}} \\ \hline \\ \rm \red{sin A} & \green{0} & \green{\dfrac{1}{2}}& \green{\dfrac{1}{ \sqrt{2} }} &\green{ \dfrac{ \sqrt{3}}{2} }&\green{1} \\ \hline \\ \rm \red{cos \: A} & \green{1} &\green{ \dfrac{ \sqrt{3} }{2}}&\green{ \dfrac{1}{ \sqrt{2} }} & \green{\dfrac{1}{2}} &\green{0} \\ \hline \\\rm \red{tan A}& \green{0} &\green{ \dfrac{1}{ \sqrt{3} }}&\green{1} & \green{\sqrt{3}} & \rm \green{\infty} \\ \hline \\ \rm \red{cosec A }& \rm \green{\infty} & \green{2}& \green{\sqrt{2} }&\green{ \dfrac{2}{ \sqrt{3} }}&\green{1} \\ \hline\\ \rm \red{sec A} & \green{1 }&\green{ \dfrac{2}{ \sqrt{3} }}& \green{\sqrt{2}} & \green{2} & \rm \green{\infty} \\ \hline \\ \rm \red{cot A }& \rm \green{\infty} & \green{\sqrt{3}}& \green{1} & \green{\dfrac{1}{ \sqrt{3} }} & \green{0}\end{array}}}}$

$\color{#FF7968}{\rule{36pt}{7pt}}\color{#4FB3F6}{\rule{36pt}{7pt}}\color{#FBBE2E}{\rule{36pt}{7pt}}\color{#60D399}{\rule{36pt}{7pt}}\color{#6D83F3}{\rule{36pt}{7pt}}$

Answer 2

Step-by-step explanation:

✞︎Given

Two poles of equal heights are standing opposite each other on either side 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 distance of the point from the poles.

✞︎To Find

Find the height of the poles and the distance of the point from the poles.

✞︎Solution

Our Answers Are -

$\tt\large\underline{\longmapsto \: Height \: Of \: The \: Pole \: Is \:20 \sqrt{3m} }$

$\tt\large\underline{\longmapsto \: Distance \: of \: the \: point \: first \: pole \: is \:  20m}$

$\huge\color{pink}\boxed{\colorbox{Black}{Second Pole=60Metre}}$

Kindly Refer To Attachment For Further Solution.

Hope It Helps!!