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Assertion (A): The shadow of a tower on a level plane is found to be 60 meters longer when sun's altitude is $30 °$ than that when it is $45 °$ .

Then the height of the tower is $30$ $(\sqrt{3} + 1)$ meters.

Reason (R): The angle of elevation of a top of a tower standing on horizontal plane from a Point $A$ is $α$ . After walking a distance $d$ meters towards the foot of the tower, the angle elevation is found to be $β$ then the height of the tower is $h = \frac{d}{cotα - cotβ}$

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Both

A

and

R

are ture and

R

is the correct explanation of

A

Both

A

and

R

are true and

R

is not correct explanation of

A

A

is true but

R

is false

A

is false but R is true

answer is A.

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Detailed Solution

It is given the shadow of a tower on a level plane is found to be 60 meters longer when sun's altitude is

30 °

. When the altitude is

45 °

, then the height of the tower is

30 (\sqrt{3} + 1)

meters.
According to the given data the figure has been drawn below.
Question Image

Let the height of the tower be

h

.
When the angle of elevation of the sun is

45 °

the height of the tower and the length of its shadow is the same.
Since in

ΔBDC,

tan θ = P e r p e n d i c u l a r b a s e

\tan 45 ° = \frac{DC}{BC}

\Rightarrow 1 = \frac{DC}{BC}

∵ tan 45 ° = 1

\Rightarrow DC = BC

Let us assume DC =

h

.
Now the elevation of the sun is

30^{0}

, so the shadow becomes the length of AC.
From the figure, AC =

h + 60

.
In

ΔDAC,

tan θ = P e r p e n d i c u l a r b a s e

\tan 30 ° = \frac{DC}{AC}

= \frac{h}{h + 60}

\Rightarrow h + 60 = hcot 30 °

∵ 1 tan θ = cot θ

\Rightarrow hcot 30 ° - h = 60

\Rightarrow h = \frac{60}{\cot 30^{0} - 1}

\Rightarrow h

= \frac{60}{\sqrt{3} - 1}

∵ cot 30 ° = 3

\Rightarrow h

= \frac{60}{2} (\sqrt{3} + 1)

(rationalizing the denominator)

\Rightarrow h

= 30 (\sqrt{3} + 1)

m
So, both

A

and

R

are ture and

R

is the correct explanation of

A

.
So, the correct option is 1.

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