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From a point on the horizontal plane, the elevation of the top of a hill is $45 ∘$ . After walking 500m in towards its summit up a slope inclined at an angle of $15 ∘$ to the horizon, the elevation is $75 ∘$ , the height of the hill is:

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90 m

48 m

2506 m

72 m

answer is C.

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Detailed Solution

Let the diagram of the situation be,
seo

Δ F D E

, we have

sin θ = P e r p e n d i c u l a r H y p o t e n u s e

sin 15 ∘ = F D E F

… (1)
We know that

sin 15 ∘ = sin (45 ∘ − 30 ∘)

, and

sin (A − B) = sin A cos B − cos A sin B

sin 15 ∘ = sin 45 ∘ cos 30 ∘ − cos 45 ∘ sin 30 ∘ = 12 ⋅ 32 − 12 ⋅ 12 = 3 − 1 22

Then, (1) becomes,

3 − 1 22 = F D E F

Given that

E F = 500 m

, and we have

3 − 1 22 = F D E F 3 − 1 22 = F D 500 F D = 3 − 1 22 × 500

F D = 250 3 − 1 2 m

… (2)
In

Δ F D E

, we have

cos θ = B a s e H y p o t e n u s e cos 15 ∘ = E D E F

Similarly, we get

cos 15 ∘ = 3 + 1 22

E D E F = 3 + 1 22

Given that

E F = 500 m

, and we have

E D 500 = 3 + 1 22 E D = 3 + 1 22 × 500

E D = 250 3 + 1 2 m

… (3)
In

Δ A C E

, we have

tan θ = P e r p e n d i c u l a r B a s e tan 45 ∘ = A C E C 1 = A C E C

A C = E C

… (4)
In

Δ A B F

, we have

tan 75 ∘ = A B B F

By using the formula

tan (A + B) = tan A + tan B 1 − tan A tan B

, we get

tan 75 ∘ = 2 + 3

. So,

Hence, the correct option is 3.

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From a point on the horizontal plane, the elevation of the top of a hill is 45 ∘ . After walking 500m in towards its summit up a slope inclined at an angle of 15 ∘ to the horizon, the elevation is 75 ∘ , the height of the hill is:

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From a point on the horizontal plane, the elevation of the top of a hill is $45 ∘$ . After walking 500m in towards its summit up a slope inclined at an angle of $15 ∘$ to the horizon, the elevation is $75 ∘$ , the height of the hill is: