Question

from the top of tower 96m high,the angles of depression of two cars on a road at the same level as base of the tower and on same side of it are theta and phi,where tan theta =3/4 and tan phi = 1/3. find the distance between two cars.

Answer 1

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Answer 2

The distance between two cars is 160 m.

Step-by-step explanation:

We are given that from the top of tower 96m high, the angles of depression of two cars on a road at the same level as the base of the tower and on the same side of it are theta and phi, where tan theta =3/4 and tan phi = 1/3.

In the figure drawn below; let $\angle ACB = \text{tan} \theta = \frac{3}{4}$  and  $\angle ADB = \text{tan } \phi = \frac{1}{3}$ and also the height of the tower = AB = 96 m.

Now, as we know that $\text{tan } \theta = \frac{\text{Perpendicular}}{\text{Base}}$

So, in $\triangle$ABC,

$\text{tan } \theta = \frac{AB}{BC}$

$\frac{3}{4 } =\frac{96}{BC}$

$BC = \frac{96 \times 4}{3}$

BC = $32 \times 4$ = 128 m

Now, in $\triangle$ABD,

$\text{tan } \phi = \frac{AB}{BD}$

$\frac{1}{3} =\frac{96}{BD}$

$BD = \frac{96 \times 3}{1}$

BD = 288 m

So, the distance between two cars = CD = BD - BC

                                                                    = 288 m - 128 m = 160 m.