Question

from the top and foot of a tower 40m high,the angle of elevation of the top of a light house is found to be 30 degree and 60 degree respectively Find the height of the light-house Also find the distance of the top of the light-house from the foot of the tower .

Answer 1

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Answer 2

Answer:

Height of the tower is 60 meters.

Distance from the top of the lighthouse from the foot of the tower is 69.3 meters.

Step-by-step explanation:

We are given that,

Height of the tower = 40 meters

Angle of elevation from the top of the tower = 30°

Angle of elevation from the bottom of the tower = 60°

Let, as shown in the figure below,

The upper part of the tower = x  meters

The distance from the top of the lighthouse to the foot = y meters.

The horizontal distance from the lighthouse to the tower = L meters

So, using the trigonometric forms for the angles, we have,

$\tan \theta=\frac{Perpendicular}{Base}$

i.e. $\tan 30=\frac{x}{L}$ and $\tan 60=\frac{40+x}{L}$

i.e. $\frac{1}{\sqrt{3}}=\frac{x}{L}$ and $\sqrt{3}=\frac{40+x}{L}$

Dividing both the terms, we get,

$\frac{1}{3}=\frac{x}{40+x}$

i.e. $40+x=3x$

i.e. $2x=40$

i.e. x = 20 meters

Thus, the height of the tower = 40 + 20 = 60 meters.

Further, we have,

$\sin \theta=\frac{Perpendicular}{Hypotenuse}$

i.e. $\sin 60=\frac{60}{y}$

i.e. $y=\frac{60}{\sin 60}$

i.e. $y=\frac{60}{0.866}$

i.e. y= 69.3 meters.

Thus, the distance from the top of the lighthouse from the foot of the tower is 69.3 meters,