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When the altitude of the Sun increases from $30 °$ to $45 °$ , the length of the shadow of a palm tree decreases by 12 m. What will be the length of the palm tree?

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63 + 2

33 + 1

6 3 + 1

62 + 1

answer is C.

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Detailed Solution

Given,
Increase in the altitude of the sun =

30 °

to 45

°

Decrease in the length of the shadow of palm tree by 12m
We draw diagram,
Question Image

We use the trigonometric identity

tan θ = p e r p e n d i c u l a r b a s e

.
We get the value of BC from

Δ A B C

tan 45 ° = A B B C ⇒ 1 = h x [∵ tan 45 ° = 1] ⇒ x = h ......... (i)

We get the value of AB from

Δ A B D

tan 30 ° = A B B D ⇒ 13 = h B C + C D [∵ tan 30 ° = 13] ⇒ 13 = h x + 12 ...... (i i)

Substituting the value of x from (i) in equation (ii),

⇒ 13 = h x + 12 ⇒ 13 = h h + 12 ⇒ h + 12 = 3 h

⇒ 12 = 3 h − h ⇒ 12 = h 3 − 1 ⇒ 12 3 − 1 = h

⇒ h = 12 3 − 1 × 3 + 1 3 + 1 ⇒ h = 12 3 + 1 3 − 1 ⇒ h = 12 3 + 1 2 ⇒ h = 6 3 + 1 m

Thus the length of palm tree will be

6 3 + 1

m.
Therefore the correct option is 3.

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