Discussion

Question:

At the foot of the mountain, the angle of elevation of the summit at the top of the mountain is 45⁰. After ascending 100 metres, at a slope of 30⁰ up the mountain towards the summit, the angle of elevation of the summit is 60⁰. Find the height of the summit.

Explanation:

Let height of the mountain = x

From the figure, AD = distance travelled by man = 100 m

Also, DE = AD × sin 30° = 100 × 0.5 = 50 m

Also, AE = AD × cos 30° = 100 × (√3/2) = 50√3 m

Now, ∠CAB = 45°

Hence, ABC is a 45-45-90 triangle and AB = BC = x

Also, EB = AB – AE = (x − 50√3) m

∴ DF = EB = (x − 50√3) m

Similarly, DE = FB = 50 m

∴ CF = BC – FB = (x – 50) m

Now, in ∆CDF, tan 60° = CF/DF

∴ √3 = (x – 50)/(x − 50√3)

∴ √3x – 50(3) = x − 50

∴ (√3 – 1)x = 150 – 50 = 100

∴ x = 100/((√3 – 1) = 50(√3 + 1) [Rationalising]

Hence, option (a).

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