Question

Q31.
One sees the top of a tree on the bank of a river at an elevation of 70° from the other bank. Stepping 20 metres back, he
sees the top of the tree at an elevation of 55°. Height of the person is 1.4 metres.
(a) Draw a rough figure and mark the measurements.
(b) Find the height of the tree.
(c) Find the width of the river.
[tan 70° = 2.75; tan 55° = 1.43]
​

Answer 1

The height of the tree is 61 m and the width of the river is 21.67 m.

Step-by-step explanation:

Referring to the figure attached below, let’s make some assumptions,

The height of the person, AB = GC = ED = 1.4 m

BC = 20 m = AG [since the person stepped 20 m back from his initial position C]

FD = height of the tree

The width of the river = CD = GE

The angle of elevation, ∠FGE = 70°

The angle of elevation after stepping 20 m back from point C, ∠FAE = 55°

Considering ∆FGE, by applying trigonometry ratios of a triangle, we have

tan θ = perpendicular/base = FE/GE

⇒ tan 70° = FE/GE  ...... [here θ = ∠FGE = 70° ]

⇒ 2.75 = FE/GE  

⇒ FE = 2.75 * GE ……. (i)

Now, considering ∆AFE, by applying trigonometry ratios of a triangle, we have

tan θ = perpendicular/base = FE/AE

⇒ tan 55° = FE/(AG+GE)   ...... [here θ = ∠FAE = 55° ]

⇒ 1.43 = FE/(20+GE)  

⇒ FE = 28.6 +1.43GE

Substituting value of FE from (i)

⇒ 2.75GE = 28.6 +1.43GE

⇒ 1.32GE = 28.6

⇒ GE = 28.6/1.32

⇒ GE = 21.67 m = CD ← width of the river ……. (ii)

Therefore, from (i) & (ii), we get

The height of the tree "FD",

= FE + ED

= (2.75 * GE) + 1.4

= (2.75*21.67) + 1.4

= 60.99

≈ 61 m

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