On the same side of a tower, two objects are located.

Question:

On the same side of a tower, two objects are located. When observed from the top of the tower, their angles of depression are 45° and 60°. If the height of the tower is 150 m,  find the distance between the objects.

Solution:

Let AB be the tower of height m and Two objects are located when top of tower are observed, makes an angle of depression from the top and bottom of tower are  and  respectively.

So we use trigonometric ratios.

In a triangle ABC,

$\tan 45^{\prime}=\frac{150}{x+y}$

$\Rightarrow x+y=150$ ............(1)

Again in a triangle ABD,

$\tan 60^{\circ}=\frac{150}{y}$

$\Rightarrow \sqrt{3}=\frac{150}{y}$]

$\Rightarrow \sqrt{3} y=150$ ........(2)

So from (1) and (2) we get

$x+\frac{150}{\sqrt{3}}=150$

$\Rightarrow \sqrt{3} x=150(\sqrt{3}-1)$

$\Rightarrow x=63.39$

Hence the required distance is approximately  m.

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