Question

a tower standing on a horizontal plane makes an angle a at a point which is 160 m apart from the foot of the tower . On moving 100 m towards the base of the tower, the angle of elevation becomes 2a. Find the height of the tower???..​

Answer 1

$\Large{\underline{\underline{\mathfrak{\red{\bf{Solution}}}}}}$

$\Large{\underline{\mathfrak{\orange{\bf{Find}}}}}$

• Height of tower

$\Large{\underline{\underline{\mathfrak{\red{\bf{Explanation}}}}}}$

Let, AB be the tower of height = x m.

and,

<BCA = a , < BDA = 2a ,

CD = 100 m , DA = 60 m

So, Now take, ∆ ABD.

➩ tan 2a = AB/AD = x/60 -------------(1)

Now, take ∆ ABC.

➩ tan a = AB/AC

➩tan a = AB/(AD+DC)

➩ tan a = x/(60+100)

➩ tan a = x/160 ------------------------(2)

Using Trigonometry Formula

★ tan 2a = 2 tan a /( 1 - tan² a)

keep value by equ(1) & equ(2)

➩ x/60 = (2 * x/160)/[ 1 - (x/160)² ]

➩ 1/60 = 160²/80*(160² - x²)

➩ 160² - x² = 2 × 160 × 60

➩ 25,600 - x² = 19,200

➩x² = 25,600 - 19,200

➩ x² = 64,00

➩ x = ±√(64,00)

➩ x = ±√(80 × 80)

➩ x = ± 80

Height be always positive .

So, keep ,

• x = 80

$\Large{\underline{\underline{\mathfrak{\red{\bf{Hence}}}}}}$

• Height of tower will be (x) = 80 meter

________________

$\Large{\underline{\mathfrak{\orange{\bf{Note}}}}}$

• For diagram see above attachment.

________________

Answer 2

$\red{ \large{Answer}}$

We know that:-

→BD= 160m

→CD= 100m

→BC= 160m -100m = 60m

__________________

$\orange{\large{ In \: \triangle ACD}}$

$\orange{ =>\angle ACB = \angle CAD +\angle AD \: (Alternative \: Angles)}$

$\orange{=>2a = \angle CAD + a}$

$\orange{ =>\angle CAD = 2a - a}$

$\orange{ =>\angle CAD = a}$

$\purple{ \therefore AC = CD}$

$\pink{ \large{ In \: \triangle ABC}}$

$\pink{=>AC = 100m}$

$\pink{ BC = 160 - 100 = 60m}$

===================

$\blue{ \underline{BY \: PYTHAGORAS \: THEOREM}}$

→H²= P² + B²

H stands for hypotenuse, P stands for perpendicular, B stands for base.

✪AC² = AB² - BC²

✪100² = AB² - 60²

✪10000-3600 = AB²

✪6400 = AB²

✪±(√6400) = AB

✪±80 = AB

$\green{ \large{ HENCE, the\: HEIGHT\: of\: the\: tower\: is :- }}$

$\red{ \boxed{ \huge{80m}}}$