Comprehension
A man standing on a boat south of a lighthouse observes his shadow to be 24 meters long, as measured at sea level. On sailing 300 meters eastwards, he finds his shadow as 30 meters long, measured in a similar manner. The height of the man is 6 meters above sea level. The height of the lighthouse above sea level is:
Question: 1
The height of the light house above the sea level is:
The height of the light house above the sea level is:
Show Hint
Always use triangle similarity when comparing heights and shadow lengths. Then apply Pythagoras to resolve distances when movement is involved.
Updated On: Aug 23, 2025
- 90 meters
- 94 meters
- 96 meters
- 100 meters
- 106 meters
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The Correct Option is
Solution and Explanation
Step 1: Define situation.
- Let the height of the lighthouse = \(KL\). - Height of man = 6 m. - Shadow length in first position = 24 m. - Shadow length in second position (after 300 m east) = 30 m.
Step 2: Use similarity of triangles.
By geometry, the lighthouse, tip of shadow, and man form similar right triangles. So, \[ \frac{KL}{24} = \frac{LE}{30} \] where \(LE\) is the horizontal distance of the man in second position from lighthouse.
Step 3: Relationship between distances.
We know that: \[ \frac{24}{30} = \frac{4}{5} \] Thus, if \(LC = 4x\), then \(LE = 5x\).
Step 4: Use Pythagoras in right triangle.
Triangle \(LBE\) is right-angled: \[ (LE)^2 = (LB)^2 + (BE)^2 \] Given: \(LB = 4x\), \(LE = 5x\), and \(BE = 300\). So, \[ (5x)^2 - (4x)^2 = 300^2 \] \[ 25x^2 - 16x^2 = 90000 \] \[ 9x^2 = 90000 \quad \Rightarrow \quad x^2 = 10000 \quad \Rightarrow \quad x = 100 \]
Step 5: Find distances.
- \(LB = 4x = 400\) m - \(LE = 5x = 500\) m - \(LC = LB + BC = 400 + 24 = 424\) m
Step 6: Find height of lighthouse.
Using similarity: \[ \frac{KL}{24} = \frac{424}{4} \] \[ KL = \frac{424}{4} \times 24 = 106 \]
Final Answer:
\[ \boxed{106 \text{ meters}} \]
- Let the height of the lighthouse = \(KL\). - Height of man = 6 m. - Shadow length in first position = 24 m. - Shadow length in second position (after 300 m east) = 30 m.
Step 2: Use similarity of triangles.
By geometry, the lighthouse, tip of shadow, and man form similar right triangles. So, \[ \frac{KL}{24} = \frac{LE}{30} \] where \(LE\) is the horizontal distance of the man in second position from lighthouse.
Step 3: Relationship between distances.
We know that: \[ \frac{24}{30} = \frac{4}{5} \] Thus, if \(LC = 4x\), then \(LE = 5x\).
Step 4: Use Pythagoras in right triangle.
Triangle \(LBE\) is right-angled: \[ (LE)^2 = (LB)^2 + (BE)^2 \] Given: \(LB = 4x\), \(LE = 5x\), and \(BE = 300\). So, \[ (5x)^2 - (4x)^2 = 300^2 \] \[ 25x^2 - 16x^2 = 90000 \] \[ 9x^2 = 90000 \quad \Rightarrow \quad x^2 = 10000 \quad \Rightarrow \quad x = 100 \]
Step 5: Find distances.
- \(LB = 4x = 400\) m - \(LE = 5x = 500\) m - \(LC = LB + BC = 400 + 24 = 424\) m
Step 6: Find height of lighthouse.
Using similarity: \[ \frac{KL}{24} = \frac{424}{4} \] \[ KL = \frac{424}{4} \times 24 = 106 \]
Final Answer:
\[ \boxed{106 \text{ meters}} \]
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Question: 2
What is the horizontal distance of the man from the lighthouse in the second position?
What is the horizontal distance of the man from the lighthouse in the second position?
Show Hint
Once the scaling ratio is found using similarity, you can directly multiply it to get distances instead of repeating the whole derivation.
Updated On: Aug 23, 2025
- 300 meters
- 400 meters
- 500 meters
- 600 meters
- None of the above
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The Correct Option is C
Solution and Explanation
Step 1: Recall from Q74.
From earlier solution, we found that: - \(x = 100\). - \(LE = 5x\).
Step 2: Compute distance.
\[ LE = 5 \times 100 = 500 \text{ meters} \]
Final Answer:
\[ \boxed{500 \text{ meters}} \]
From earlier solution, we found that: - \(x = 100\). - \(LE = 5x\).
Step 2: Compute distance.
\[ LE = 5 \times 100 = 500 \text{ meters} \]
Final Answer:
\[ \boxed{500 \text{ meters}} \]
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