Question

A distant cliff has 10degree and 12 degree angles of elevation at two places as 100 m apart what is height of cliff

Answer 1

Given:

A distant cliff has 10° and 12° angles of elevation at two places as 100 m apart

To find:

The height of the cliff

Solution:

From the figure attached below, we have

AD = height of the cliff

BC = distance between the two places = 100 m

∠ABC = 10° = angle of elevation from point B to the top of the cliff

∠ACB = 12° = angle of elevation from point C to the top of the cliff

Formula to be used:

Trigonometric ratio of a triangle:- tan θ = $\frac{Perpendicular}{Base}$

Let's assume,

"x" meters → be the distance between B and D

"100 - x" meters → be the distance between C and D

Consider ΔABD, we have

AD = perpendicular

BD = base

θ = 10°

∴ tan 10° = $\frac{AD}{BD}$

⇒ tan 10° = $\frac{AD}{x}$

⇒ 0.176x = AD ....... (i)

Consider ΔADC, we have

AD = perpendicular

CD = base

θ = 12°

∴ tan 12° = $\frac{AD}{CD}$

⇒ tan 12° = $\frac{AD}{100 - x}$

⇒ 0.212(100 - x) = AD ....... (ii)

Now, comparing eq. (i) & (ii), we get

0.176x = 0.212(100 - x)

⇒ 0.176x = 21.2 - 0.212x

⇒ 0.176x + 0.212x = 21.2

⇒ 0.388x = 21.2

⇒ x = $\frac{21.2}{0.388}$

⇒ x = 54.63 m

By substituting the value of x in eq.(i), we get

AD =  0.176x = 0.176 * 54.63 = 9.61 m

Thus, the height of the cliff is 9.61 m.

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