A straight highway leads to the foot of a tower of height $ 50 \mathrm{~m} $. From the top of the tower, the angles of depression of two cars standing on the highway are $ 30^{\circ} $ and $ 60^{\circ} $ respectively. What is the distance between the two cars and how far is each car from the tower?

Given:

A straight highway leads to the foot of a tower of height $ 50 \mathrm{~m} $.

From the top of the tower, the angles of depression of two cars standing on the highway are $ 30^{\circ} $ and $ 60^{\circ} $ respectively.

To do:

We have to find the distance between the two cars and the distance between each car from the tower.

Solution:

Let $AB$ be the height of the tower and $C, D$ be the points where the two cars are whose angles of depression are $ 30^{\circ} $ and $ 60^{\circ} $ respectively.

From the figure,

$\mathrm{AB}=50 \mathrm{~m}, \angle \mathrm{BCA}=30^{\circ}, \angle \mathrm{BDA}=60^{\circ}$

Let the car $C$ be at a distance of $\mathrm{AC}=x \mathrm{~m}$ from the foot of the tower and the distance between the cars $C$ and $D$ be $\mathrm{CD}=y \mathrm{~m}$.

This implies,

$\mathrm{DA}=x-y \mathrm{~m}$

We know that,

$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$

$=\frac{\text { AB }}{AC}$

$\Rightarrow \tan 30^{\circ}=\frac{50}{x}$

$\Rightarrow \frac{1}{\sqrt3}=\frac{50}{x}$

$\Rightarrow x=50\sqrt3=50(1.73)=86.5 \mathrm{~m}$..........(i)

Similarly,

$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$

$=\frac{\text { AB }}{DA}$

$\Rightarrow \tan 60^{\circ}=\frac{50}{x-y}$

$\Rightarrow \sqrt3=\frac{50}{50\sqrt3-y}$ [From (i)]

$\Rightarrow (50\sqrt3-y)\sqrt3=50 \mathrm{~m}$

$\Rightarrow 50(3)-y\sqrt3=50 \mathrm{~m}$

$\Rightarrow y\sqrt3=150-50 \mathrm{~m}$

$\Rightarrow y=\frac{100}{1.73} \mathrm{~m}$

$\Rightarrow y=57.67 \mathrm{~m}$

$\Rightarrow x-y=50(1.73)-57.67=86.6-57.67=28.83 \mathrm{~m}$

Therefore, the distance between the two cars is $57.67 \mathrm{~m}$, the distance between each car and the tower is $86.6 \mathrm{~m}$ and $28.83 \mathrm{~m}$.

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Updated on: 10-Oct-2022

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