Question

a pole of length 10cm costs a shadow 8m long on the ground .at the same time a tower cases a shadow of length 50 m on the ground then find the height of the tower​

Answer 1

Correction in the question: A pole of length 10m costs a shadow 8m long on the ground. At the same time, a tower casts a shadow of length 50 m on the ground. Then find the height of the tower​ .

Solution:

Pole = 10m = AB

Shadow cast by AB = 8m = BC

Tower = x meters = PQ

Shadow cast by PQ = 50m = QR

Now, in ΔABC & ΔPQR.

→ ∠ABC = ∠PQR = 90° (Tower & Pole are perpendicular to the ground)

→ ∠ACB = ∠PRQ (The shadows are cast at the same time, hence the angle of elevation of the top of tower & pole will be the same)

∴ By applying AA similarity criterion we get:

⇒ ΔABC ≈ ΔPQR.

We know that if two triangles are similar, the ratio of their corresponding sides are in proportion. Therefore:

$\rm \Longrightarrow \dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{CA}{RP}$

$\rm \Longrightarrow \dfrac{AB}{PQ} = \dfrac{BC}{QR}$

$\rm \Longrightarrow \dfrac{10}{x} = \dfrac{8}{50}$

$\rm \Longrightarrow x = \dfrac{50 \times 10}{8}$

$\rm \Longrightarrow x = \dfrac{500}{8}$

$\rm \Longrightarrow x = 62.5$

Therefore the height of the tower is 62.5m.

Answer 2

[tex]Correction in the question: A pole of length 10m costs a shadow 8m long on the ground. At the same time, a tower casts a shadow of length 50 m on the ground. Then find the height of the tower.

Solution:

Pole = 10m = AB

Shadow cast by AB = 8m = BC

Tower = x meters = PQ

Shadow cast by PQ = 50m = QR

Now, in ΔABC & ΔPQR.

→ ∠ABC = ∠PQR = 90° (Tower & Pole are perpendicular to the ground)

→ ∠ACB = ∠PRQ (The shadows are cast at the same time, hence the angle of elevation of the top of tower & pole will be the same)

∴ By applying AA similarity criterion we get:

⇒ ΔABC ≈ ΔPQR.

We know that if two triangles are similar, the ratio of their corresponding sides are in proportion. Therefore:

\rm \Longrightarrow \dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{CA}{RP}⟹PQAB=QRBC=RPCA

\rm \Longrightarrow \dfrac{AB}{PQ} = \dfrac{BC}{QR}⟹PQAB=QRBC

{10}{x} = \dfrac{8}{50}⟹x10=508

\rm \Longrightarrow x = \dfrac{50 \times 10}{8}⟹x=850×10

\rm \Longrightarrow x = \dfrac{500}{8}⟹x=8500

\rm \Longrightarrow x = 62.5⟹x=62.5

Therefore the height of the tower is 62.5m.