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
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Important Questions from Heights and Distances
Two pillars are placed vertically 8 feet apart. The height difference of the two pillars is 6 feet. The two ends of a rope of length 15 feet are tied to the tips of the two pillars. The portion of the length of the taller pillar that can be brought in contact with the rope without detaching the rope from the pillars is
- 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?
- In a right angled triangle, right-angled at $A$, the angle formed at $B$ is $45^\circ$, if the length of side $AB$ is 3m, then what is the length of the side $BC$?
