Question

3. When one looks from the foot and the top of a tower from the roof of a building the angles of
elevation and depression are of 63 and 27° respectively. If the height of the building is 20
metres, find the height of the tower. (tan 63° = 2)​

Answer 1

Solution

→ tan 63° = BC/CD

→ 2 = 20/CD

→ CD = 20/2 → CD = 10 m

Now, cot(90° - ∅) = tan∅

→ cot(90° - 63)° = tan 63°

→ cot 27° = tan 63° = 2

→ cot 27° = BE/AE

→ 2 = 10/AE [Because BE = CD]

→ AE = 10/2 → AE = 5 m

Now, BC = DE = 20 m

→ AD = AE + DE

→ AD = 5 m + 20 m

→ AD = 25 m

Hence height of tower is 25 m.

Answer 2

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

Height of the building = 25 m.

$\bold{\underline{\underline{\large{\sf{StEp\:by\:stEp\:explanation:}}}}}$

GiVeN :

• Angle of elevation = 27°
• Angle of depression = 63°
• Height of the building = 20 m.

To FiNd :

• Height of the tower.

SoLuTiOn :

Let PS = height of the tower = x metre.

In Δ RTS,

RT = height of the building = 20 m

m $\angle{RST}$ = 63°

Let $\theta$ = 63°

We have opposite side to angle $\theta$ and value of tan 63° = 2

•°• We can use tan to find the base.

$\bold{tan\theta\:=\:{\dfrac{Opposite\:side\:to\:\angle\:\theta}{Adjacent\:side\:to\:\angle\:\theta}}}$

$\longrightarrow$$\bold{tan\theta\:=\:{\dfrac{RT}{ST}}}$

$\longrightarrow$$\bold{tan\:63\degree\:=\:{\dfrac{20}{ST}}}$

$\longrightarrow$$\bold{2\:=\:{\dfrac{20}{ST}}}$

$\longrightarrow$$\bold{ST={\dfrac{20}{2}}}$

$\longrightarrow$$\bold{ST=10}$

•°• Base = ST = 10 m.

ST = QR = 10 m

Now,

$\longrightarrow$cot (90 - $\theta$ ) = tan $\theta$

$\longrightarrow$cot ( 90° - 63°) = tan 63°

$\longrightarrow$cot 27° = tan 63°

We have been given tan 63° = 2

•°• Cot 27 = 2.

Let $\theta$ = 27°

Cot $\theta$ = $\bold{\dfrac{Adjacent\:side\:to\:\angle\:\theta}{Opposite\:side\:to\:\angle\:\theta}}$

Block in the values,

$\longrightarrow$2 = $\bold{\dfrac{RQ}{PQ}}$

$\longrightarrow$2 = $\bold{\dfrac{10}{PQ}}$

$\longrightarrow$$\bold{PQ={\dfrac{10}{2}}}$

$\longrightarrow$$\bold{PQ=5\:m}$

Now, we can find the height of the building.

$\longrightarrow$QS = RT = 20 m

$\longrightarrow$PQ = 5 m

Height of the building,

$\longrightarrow$PS = QS + PQ

$\longrightarrow$PS = 20 + 5

$\longrightarrow$PS = 25 m.

•°• Height of the building is 25 m.