Question

A man looks from the top of a vertical tower 30 metres high at a marked point upon the horizontal plane on which the tower stands. The angle of depression of this point is 30°. Find the distance of the marked point form the foot of the tower.​

Answer 1

We've triangle, ∆POM.

And, Top of a vertical tower 30 metres high at a marked point upon the horizontal plane on which the tower stands.

☯ Let's consider, OM is the height of tower.

Angle of depression, $\sf \angle P = 30^{\circ}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

Now,

$\implies\sf \dfrac{OM}{PO} = \dfrac{Perpendicular}{Base}\\\\\\:\implies\sf\dfrac{OM}{PO} = tan \ 30^{\circ}\\\\\\:\implies\sf\dfrac{30}{PO} = \dfrac{1}{\sqrt{3}}\\\\\\:\implies\boxed{\frak{\pink{PO = 30\sqrt{3} \ m}}}$

⠀⠀⠀⠀$\therefore\:{\underline{\sf{Hence,\:The \ distance \ of \ marked \ point \ from \ foot \ of \ tower \ is \ \bf{ 30\sqrt{3}}.}}}$⠀⠀

Answer 2

${\large{\bold{\sf{\underbrace{Understanding \; the \; question}}}}}$

➨ This question says that there is a man and he look from the top of a vertical tower 30 metres high at a marked point upon the horizontal plane on which the tower stands! And the angle of depression of this point is 30°. Now at last we have to find the distance of the marked point form the foot of the tower.

${\large{\bold{\sf{\underline{Given \; that}}}}}$

➨ A man look from the top of a vertical tower 30 metres high at a marked point upon the horizontal plane on which the tower stands. ( Height 30 m )

➨ The angle of depression of this point is 30°.

${\large{\bold{\sf{\underline{To \; find}}}}}$

➨ The distance of the marked point form the foot of the tower.

${\large{\bold{\sf{\underline{Solution}}}}}$

➨ The distance of the marked point form the foot of the tower = 30√3 m

${\large{\bold{\sf{\underline{Assumption}}}}}$

➨ Triangle ABC is given here.

➨ BC is the height.

➨ Angle of depression = <A

${\large{\bold{\sf{\underline{Full \; solution}}}}}$

~ According to the question,

$\sf In \: triangle \: ABC \begin{cases} & \sf{BC \: is \: the \: height = \bf{30 \: m}} \\ & \sf{A \: is \: angle \: of \: depression = \bf{30 \degree}} \end{cases}$

~ Now again according to the question,

➨ ${\bold{\sf{\dfrac{BC}{AB}}}}$ =

${\bold{\sf{\dfrac{B}{P}}}}$

Where,

➨ B denote Base

➨ P denotes Perpendicular

~ Now, as we already know that

➨ ${\bold{\sf{\dfrac{B}{P}}}}$ = tan30°

• Cross multiplying the digits

➨ ${\bold{\sf{\dfrac{30}{AB}}}}$ = ${\bold{\sf{\dfrac{1}{\sqrt{3}}}}}$

• Cross multiplying the digits again

➨ 30 × √3 = AB × 1

➨ 30√3 = AB

➨ AB = 30√3 m

• Henceforth, the distance of the marked point form the foot of the tower = 30√3 m

${\large{\bold{\sf{\underline{Additional \: knowledge}}}}}$

Height and distance -

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Distance and Height -

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Triangle -

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Right angle Triangle -

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Request : Please see the attachment too to understand the concept easily. Please see this answer from web browser or chrome just saying because I give some diagrams here but they are not shown in app.

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