Question

Question !

The angle of the elevation of the top of a tower is 30°. If the height of the tower is doubled .Then prove that the angle of elevation of its top will also be doubled .
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Answer 1

Solution

Given:-

$\sf :\implies \: Angle \: of \: elevation \: = 30^{o}$

Let

$\sf : \implies \: Height \: of \: tower \: = h \: \: \: and \: \: distance \: = \: x \:$

$\sf \implies \: \tan30 \degree = \dfrac{h}{x} \: \: \: \: \: \: \: where \: \: tan30 \degree = \dfrac{1}{ \sqrt{3} }$

By putting the value we get

$\sf \implies \: \dfrac{1}{ \sqrt{3} } = \dfrac{h}{x}$

Now height is doubled = 2h

$\sf \implies \: \tan\theta = \dfrac{2h}{x}$

Its given

$\sf \implies \: \dfrac{h}{x} = \dfrac{1}{ \sqrt{3} } \: \: \: so \: put \: the \: value \:$

we get

$\sf \implies \: \tan\theta = \dfrac{2}{ \sqrt{3} }$

When Θ is double

$\sf \implies \: \theta = 30 \degree \: \implies2 \theta = 2 \times 30 = 60$

now we know that

$\sf \implies \: \tan60 \degree = \sqrt{3} \implies \sqrt{3} \not = \dfrac{2}{ \sqrt{3} }$

So our conclusion is when height is double Θ will be not doubled

Answer 2

The angle of elevation at the top of a tower is - 30°

Let the height of the tower be - x unit

Now the base of the tower will be -

tan 30° = height / base

➡ base = height / tan 30°

➡ base = x / tan 30° ______(i)

If the height of the tower is doubled then,

Height' = 2x unit

Let the angle be tan y°

Base will remain same as only the height is doubled but the base is same so,

Base = height / tan y°

➡ Base = 2x / tan y° _____(ii)

After equalizing (i) and (ii) we get,

x / tan 30° = 2x / tan y°

➡ tan y° × x = 2x × tan 30° [ By cross multiplying ]

➡ tan y° = ( 2x × tan 30° ) / x

➡ tan y° = 2 × tan 30°

Therefore,

Proved that if the height is doubled then the angle will also get doubled.

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$\huge\mathrm{Notes}$

★ You can take the height h or any other character also ( in place of x )

★ Since the unit is not mentioned in the question so you must write " unit " only after x, 2x and not metre/ cm / km

★ The diagram should be simply like a right angled triangle. Take the height or perpendicular as the height ofthe tower and the base as the base of tower . The angle where the hypotenuse side and the base side is meeting is the angke of elevation

★ Tan theta = Height / Base