Question

from the top of a building 12m high, the angle of elevation of the top of a tower is found to
be 30 degrees.from the bottom of the same building the angle of elevation of the top of the tower
is found to be 60°. Determine the height of the tower and the distance between the tower
and building.
​

Answer 1

Solution

• AB = Building = 12 m
• CD = Tower

→ tan 30° = DE/AE

→ 1/√3 = DE/AE

→ AE = √3 DE

→ tan 60° = CD/BC

→ √3 = (CE + DE)/AE

→ √3 = (12 + DE)/√3 DE

→ 3 DE = 12 + DE

→ 2 DE = 12

→ DE = 6m

→ AE = 6√3 m or 10.392 (distance between building and tower)

→ √3 = CD/BC → √3 = CD/6√3

→ CD = 18 m (length of tower)

Answer: 18 m as length of tower & 10.392 as distance between tower & building.

Answer 2

$\bold{\underline{\underline{\huge{\sf{\red{AnsWer:}}}}}}$

Given :

• Height of the building = DC = 12 m
• Angle of elevation to the top of tower = 30°
• Angle of elevation from bottom of the building to the top of the tower = 60°

To Find :

• The height of the tower (EB)
• Distance between the tower
• between the towerand building (CB)

Solution :

In Δ EDA,

• $\theta$ = 30°
• AD = Adjacent side to $\theta$
• AE = Opposite side to $\theta$

Using Trigonometric ratio Tan,

Let angle = $\theta$

$\mathtt{Tan\:\theta\:=\:{Opposite\:side\:to\:\theta}{Adjacent\:side\:to\:theta}}$

Block in the given data,

➟ $\mathtt{Tan\:30\:\degree\:=\:{\dfrac{AE}{AD}}}$

➟ $\mathtt{\dfrac{1}{\sqrt3}}$ = $\mathtt{\dfrac{AE}{AD}}$

➟ $\mathtt{AD\:=\:{\sqrt{3}\:AE}}$ ---> (i)

In Δ ECB,

• $\theta$ = 60°
• CB = Adjacent side to $\theta$
• EB = Opposite side to $\theta$

Again using the trigonometric ratio,Tan.

➟ $\mathtt{Tan\:60\:\degree\:=\:{\dfrac{EB}{CB}}}$

➟ $\mathtt{\sqrt{3}\:=\:{\dfrac{EA\:+\:AB}{CB}}}$

$\sf{\underbrace{EB\:=\:EA\:+\:AB,\:E-A-B}}$

➟ $\mathtt{\sqrt{3}\:=\:{\dfrac{EA\:+\:12}{CB}}}$

➟ $\mathtt{\sqrt{3}\:=\:{\dfrac{EA\:+\:12}{AD}}}$

➟ $\mathtt{\sqrt{3}\:=\:{\dfrac{EA+\:12}{\sqrt{3}\:AE}}}$ ----> From (i)

➟ $\mathtt{\sqrt{3}\:\times\:\sqrt{3}\:AE\:=\:EA\:+\:12}$

➟ $\mathtt{3AE\:=\:AE+12}$

➟ $\mathtt{3AE\:-\:AE\:=\:12}$

➟ $\mathtt{2AE\:=\:12}$

➟ $\mathtt{AE\:=\:{\dfrac{12}{2}}}$

➟ $\mathtt{AE\:=\:6\:m}$

Height of the tower EB :

➟ $\mathtt{EB\:=\:EA\:+\:AB}$

➟ $\mathtt{EB\:=\:6\:m\:+\:12\:m}$

➟ $\mathtt{EB\:=\:18\:m}$

Distance between the tower and building AD :

➟ $\mathtt{\sqrt{3}\:DE\:=\:AD}$ ---- Using (i)

➟ $\mathtt{\sqrt{3}\:\times\:6\:=\:AD}$

➟ $\mathtt{1.73\:\times\:6\:=\:AD}$

➟ $\mathtt{10.38\:m\:=\:AD}$

$\bold{\large{\boxed{\tt{Height\:of\:the\:tower\:=\:EB\:=\:18\:m}}}}$

$\bold{\large{\boxed{\tt{Distance\:between\:tower\:and\:building\:=\:AD\:=\:10.38\:m}}}}$