Question

The angle of elevation of the top of a vertical tower from a point on the ground is 60 degree. From another point 10m vertically above the first, it's angle of elevation is 45 degree. Find the height of the tower.

Answer 1

(*base = b, h = height.) total height of tower = 10 +x
Applying trigonometry in the triangle with angle 45,
tan 45 = x/b (as both bases will be same)
tan 45 = 1, 1 = x/b 
x = b (cross multiplication)
now in the triangle with angle 60, 
tan 60 = total h/b (total h = x +10)
root 3 = 10+x/b
b(root3) = 10 + b
b (root3 - 1) = 10
b = 10/root3 - 1 
root 3 = 1.73
10/1.73-1 = b
10/0.73 = b
b = 13.69 (approx)
total h = x + 10
x = b
total h = b+10 = 13.69 + 10 = 23.69

Answer 2

Step-by-step explanation:

Let the height be EC

Let the the angle EAD be 45° and the angle EBC be 60°

Let EB be x m and in reactangle ABCD we know that opposite sides are equal: AB = CD = 10 m and BC = AD

Now,

$\sf In \: \Delta \: AED,$

$:\implies \sf tan \: 45^{\circ} = \dfrac{ED}{AD}$

$:\implies \sf tan \: 45^{\circ} = \dfrac{ED}{BC}$

$:\implies \sf tan \: 45^{\circ} = \dfrac{x}{BC}$

$:\implies \sf 1= \dfrac{x}{BC}$

$:\implies \sf x = BC \:\: \: \: \:\:\:\Bigg\lgroup \bf Equation\:(i)\Bigg\rgroup$

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$\sf In \: \Delta \: EBC,$

$:\implies \sf tan \: 60^{\circ} = \dfrac{EC}{BC}$

$:\implies \sf \sqrt{3} = \dfrac{x + 10}{BC}$

$:\implies \sf BC (\sqrt{3} ) = x + 10 \:\: \: \: \:\:\:\Bigg\lgroup \bf taking\:BC \: towards \:LHS \Bigg\rgroup$

$:\implies \sf x\sqrt{3} - x = 10$

$:\implies \sf x \: \left(\sqrt{3} - 1 \right) = 10$

$:\implies \sf x = \dfrac{10}{( \sqrt{3} - 1) } \:\: \: \: \:\:\:\Bigg\lgroup \bf Equation\:(ii)\Bigg\rgroup$

_______________________

★ Height of the tower will be :

$\dashrightarrow\:\: \sf Height= x+ 10$

$\dashrightarrow\:\: \sf Height= \frac{10}{ \sqrt{3 } - 1} + 10$

$\dashrightarrow\:\: \sf Height= \dfrac{10 \sqrt{3}}{ \sqrt{3} - 1 }$

$\dashrightarrow\:\: \sf Height=15 + 5 \sqrt{3}$

$\dashrightarrow\:\: \sf Height=15 + 5 \times 1.73 \: \: \: \Bigg\lgroup \bf Putting \: \sqrt{3} = 1.73 \Bigg\rgroup$

$\dashrightarrow\:\: \sf Height=15 + 8.65$

$\dashrightarrow\:\: \underline{ \boxed{ \sf Height=23.65 \: m}}$

⠀

$\therefore \: \underline{\textsf{The height of the tower is \textbf{23.65 m}}}.$