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 ${45}^{\circ }and{60}^{\circ }$. If the height of the tower is 150 m, find the distance between the objects.

Answer 1

Let AB be the tower of height 150m and Two objects are located when top of tower are observed, makes an angle of depression from the top and bottom of tower are 45 and 60 respectively.

Let CD = x, BD = y and $\angle ADB=60,\angle ACB=45$

So we use trigonometric ratios.

In a triangle ABC,

$tan45=\frac{150}{x+y}$

$x+y=150$ ------(1)

Again in triangle ABD,

$tan60=\frac{150}{y}$

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

From (1)&(2)

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

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

$x=63.39$

Hence the required distance is approximately 63.4 m.