Question:
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 :
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 $30^\circ$.
- 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
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
The height of a tower is given as $30\text{ m}$.
At a specific instant, a car is at a horizontal distance of $10\sqrt{3}\text{ m}$ from the base of this tower.
We need to find the angle of elevation of the top of the tower from the car at this position.
Step 2: Key Formula or Approach:
This problem can be represented as a right-angled triangle.
Let the height of the tower be the perpendicular ($AB = 30\text{ m}$).
Let the horizontal distance from the base of the tower to the car be the base ($BC = 10\sqrt{3}\text{ m}$).
Let the angle of elevation be $\theta$.
The trigonometric ratio relating the perpendicular and the base is tangent:
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} \]
Step 3: Detailed Explanation:
• 1. Identify the given values:
- Height of the tower (Perpendicular), $h = 30\text{ m}$
- Distance of the car from the base (Base), $x = 10\sqrt{3}\text{ m}$
• 2. Formulate the equation using the tangent ratio:
\[ \tan \theta = \frac{h}{x} \]
• 3. Substitute the values of $h$ and $x$ into the equation:
\[ \tan \theta = \frac{30}{10\sqrt{3}} \]
• 4. Simplify the fraction by canceling common factors:
\[ \tan \theta = \frac{3}{\sqrt{3}} \]
• 5. Rationalize the denominator or rewrite 3 as $(\sqrt{3})^2$:
\[ \tan \theta = \frac{(\sqrt{3})^2}{\sqrt{3}} = \sqrt{3} \]
• 6. Find the angle $\theta$ for which $\tan \theta = \sqrt{3}$:
We know from standard trigonometric values that:
\[ \tan 60^\circ = \sqrt{3} \]
Therefore:
\[ \theta = 60^\circ \]
Step 4: Final Answer:
The angle of elevation of the top of the tower from the car is $60^\circ$, which corresponds to option (D).
The height of a tower is given as $30\text{ m}$.
At a specific instant, a car is at a horizontal distance of $10\sqrt{3}\text{ m}$ from the base of this tower.
We need to find the angle of elevation of the top of the tower from the car at this position.
Step 2: Key Formula or Approach:
This problem can be represented as a right-angled triangle.
Let the height of the tower be the perpendicular ($AB = 30\text{ m}$).
Let the horizontal distance from the base of the tower to the car be the base ($BC = 10\sqrt{3}\text{ m}$).
Let the angle of elevation be $\theta$.
The trigonometric ratio relating the perpendicular and the base is tangent:
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} \]
Step 3: Detailed Explanation:
• 1. Identify the given values:
- Height of the tower (Perpendicular), $h = 30\text{ m}$
- Distance of the car from the base (Base), $x = 10\sqrt{3}\text{ m}$
• 2. Formulate the equation using the tangent ratio:
\[ \tan \theta = \frac{h}{x} \]
• 3. Substitute the values of $h$ and $x$ into the equation:
\[ \tan \theta = \frac{30}{10\sqrt{3}} \]
• 4. Simplify the fraction by canceling common factors:
\[ \tan \theta = \frac{3}{\sqrt{3}} \]
• 5. Rationalize the denominator or rewrite 3 as $(\sqrt{3})^2$:
\[ \tan \theta = \frac{(\sqrt{3})^2}{\sqrt{3}} = \sqrt{3} \]
• 6. Find the angle $\theta$ for which $\tan \theta = \sqrt{3}$:
We know from standard trigonometric values that:
\[ \tan 60^\circ = \sqrt{3} \]
Therefore:
\[ \theta = 60^\circ \]
Step 4: Final Answer:
The angle of elevation of the top of the tower from the car is $60^\circ$, which corresponds to option (D).
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