Understanding the Geometry
The problem involves a right-angled triangle formed by the candle, its shadow on the table, and the line from the table surface to the candle's flame tip. The candle represents the height (opposite side), the shadow represents the base (adjacent side), and the angle of elevation is given.
Information Provided
- Height of the candle: \( H = 21 \text{ cm} \)
- Angle of elevation: \( \theta = 60^\circ \)
- We need to find the length of the shadow on the table (distance from the candle base), denoted as \( B \).
Applying Trigonometric Concepts
In a right-angled triangle, the tangent function relates the angle of elevation to the opposite and adjacent sides:
$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $Substituting the known values:
$ \tan(60^\circ) = \frac{H}{B} $Step-by-Step Calculation
- Plug in the values: $ \tan(60^\circ) = \frac{21 \text{ cm}}{B} $
- Use the known value for \( \tan(60^\circ) \), which is \( \sqrt{3} \): $ \sqrt{3} = \frac{21 \text{ cm}}{B} $
- Rearrange the equation to solve for \( B \): $ B = \frac{21 \text{ cm}}{\sqrt{3}} $
- Simplify the expression by rationalizing the denominator: $ B = \frac{21 \text{ cm}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{21\sqrt{3}}{3} \text{ cm} $
- Further simplify the result: $ B = 7\sqrt{3} \text{ cm} $
Conclusion
The calculated length of the shadow is \( 7\sqrt{3} \text{ cm} \). Comparing this numerical value to the given options, which are in meters, we find that the value \( 7\sqrt{3} \) matches Option D.
Similar Questions
- A 1.8 m tall boy is standing at some distance from a 12 m tall building. The angle of elevation from his eyes to the top of the building increases from \(45^\circ\) to \(60^\circ\) as he walks towards the building. Find the distance (in meters) he walked towards the building.
- A tree breaks due to storm and the broken part bends, so that the top of the tree touches the ground making an angle of \(30^\circ\) with the ground. The distance between the foot of the tree to the point where the top touches the ground is 18 m. find the height of the tree (in metres).
- A 165cm tall man standing 500cm away from the wall looks at a hook on the wall with an angle of elevation of $45^\circ$. What is the height at which the hook is in the wall?
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- The angle of elevation of the sun, when the length of the shadow of a tree is equal to the height of the tree, is:
- The angle of elevation of a ladder leaning against a wall is \(45^\circ\) and the foot of the ladder is 15m away from the wall. The length of the ladder is:
Important Questions from Heights and Distances
A peacock sitting at the top of a 3 meter high pole saw a snake approaching towards pole at a distance three times of the height of the pole. Then it jumping from pole will catch the snake at what distance from the pole if both are running with same speed ?
The foot of a ladder 25 m long is 7 m from the base of the building. If the top of the ladder slips by 4 m, then by how much distance will the foot of the ladder slide?
Two hotels stand 25 m apart. One of them is 70 m high and the angle of depression of the top of other as observed from the top of this hotel is 45°. Height of the other hotel is:
If the angles of elevation of a balloon from two consecutive kilometer-stones along a straight road are 30° and 60° respectively, then the height of the balloon above the ground will be:
The angle of elevation of the top of a tower from a point 20 m away from its base is 45 °. What is the height of the tower?
