Question

A housefly is sitting on top of the lamp and was looking at the dustbin which was 20m high. the angle of depression from the top and the bottom of the dustbin from the housefly position where 30° and 45° respectively.

What is hight of lamp ?

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Answer 1

Given :

• Height of dustbin = 20m

• The angle of depression from the top and the bottom of the dustbin from the housefly position where 30° and 45° respectively.

To find :

• Height of lamp

Concept used :

• tan 45° = 1

• tan 30° = 1/√3

• tan x = perpendicular / base

Solution :

Explanation of diagram :-

• BC = dustbin of 20m

• AD = lamp

• CD = distance between lamp and dustbin

Now, in triangle ACD :-

tan 45° = AD/CD

1 = AD/CD

CD = AD

Let AE be x :-

therefore AD = x + 20

CD =x + 20

Now, in triangle ACD :-

tan 30° = AE / BE

Now , AE = x

BE = CD

CD = AD

therefore , BE = CD = AD

tan 30° = x / x + 20

1/√3 = x / (x + 20)

Now cross multiply :-

x+20= √3x

20= √3x

20= √3x - x

20= √3x - x

value of √3= 1.73

20= 1.73 x - x

20= 0.73 x

20 / 0.73 = x

x = 27.39 m

AD = AE + ED

AD = x+ 20

AD = 27.39 + 20

AD = 47.39 m

Answer :

height of lamp = 47.39 m

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Learn more:-

Sin 30° = 1/2

cos 30° = √3/2

tan 30° = 1/√3

Sin 45° = 1/√2

cos 45° = 1/√2

tan 45° = 1

Sin 60° = √3/2

cos 60° = 1/2

tan 60° = √3

Sin 90° = 1

cos 90° = 0

tan 90° = infinite

cosec x = 1/ sin x

sec x = 1/ cosx

cot x = 1/tan x

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Answer 2

Given ,

The height of lamp = 20 m

The angle of depression from the top and the bottom of the dustbin from the housefly position are 30 and 45

In Δ ABC ,

$\tt \implies \tan(30) = \frac{h}{BC}$

$\tt \implies \frac{1}{ \sqrt{3} } = \frac{h}{BC}$

$\tt \implies BC = \sqrt{3} h$

Now , in Δ ADE ,

$\tt \implies \tan(45) = \frac{h + 20}{DE}$

$\tt \implies1 = \frac{h + 20}{DE}$

$\tt \implies DE = h + 20$

Since , BC = DE

Thus ,

$\tt \implies h \sqrt{3} = h + 20$

$\tt \implies h( \sqrt{3} - 1) = 20$

$\tt \implies h = \frac{20}{ \sqrt{3} - 1}$

$\tt \implies h = \frac{20}{0.732}$

$\tt \implies h = 27.3$

Since , height of lamp = (20 + h) meter

Thus ,

Height of lamp = 20 + 27.3

Height of lamp = 47.3 meter

Therefore , the height of lamp is 47.3 m

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