Question:medium

A car is moving away from the base of a 30 m high tower. The angle of elevation of the top of the tower from the car at an instant, when the car is $10\sqrt{3}$ m away from the base of the tower, is :

Show Hint

Remember the two highly common ratios in height and distance problems:
- If the height is $\sqrt{3}$ times the base, the angle of elevation is $60^\circ$.
- If the base is $\sqrt{3}$ times the height, the angle of elevation is $30^\circ$.
Updated On: Jul 7, 2026
  • $30^\circ$
  • $45^\circ$
  • $90^\circ$
  • $60^\circ$
Show Solution

The Correct Option is D

Solution and Explanation

Step 1: Recognise the special right-triangle ratio.
Rather than computing $\tan\theta$ and then looking up the angle, notice the given numbers directly match the classic $30^{\circ}\text{-}60^{\circ}\text{-}90^{\circ}$ triangle, whose sides are always in the fixed ratio $1 : \sqrt{3} : 2$ opposite the angles $30^{\circ}, 60^{\circ}, 90^{\circ}$.

Step 2: Match the given sides to this ratio.
The base (distance of car from tower) is $10\sqrt{3}\text{ m}$ and the height (perpendicular) is $30\text{ m}$. Write both in terms of a common scale factor $k$:
\[ 10\sqrt{3} = k\sqrt{3} \implies k = 10 \] \[ 30 = 3k = 3(10) = 30 \] Both sides scale consistently with $k = 10$ against the ratio parts $\sqrt{3}$ and $3$ (which is $\sqrt{3} \times \sqrt{3}$, corresponding to a triangle where the side opposite $60^{\circ}$ is $\sqrt{3}$ times the side opposite $30^{\circ}$).

Step 3: Identify which angle is opposite the height.
In this right triangle, the height $30\text{ m}$ (the longer leg) is opposite the angle of elevation $\theta$, and the base $10\sqrt{3}\text{ m}$ (the shorter leg) is opposite the remaining acute angle. Since the longer leg is $\sqrt{3}$ times the shorter leg here
\[ \frac{30}{10\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \] this matches the leg-ratio pattern of a $30^{\circ}\text{-}60^{\circ}\text{-}90^{\circ}$ triangle where the angle opposite the longer leg is $60^{\circ}$.

Step 4: State the angle.
So the angle of elevation $\theta = 60^{\circ}$.

Final Answer:
The angle of elevation of the top of the tower from the car is $60^{\circ}$, which corresponds to option (D). \[ \boxed{60^{\circ}} \]
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