A vertical cylindrical container is filled with oil. A solid hemispherical stone of radius 7 cm is immersed completely, and the o…

Question

A vertical cylindrical container is filled with oil. A solid hemispherical stone of radius 7 cm is immersed completely, and the oil level rises by 3 cm. What is the radius of the cylinder?

1. A. 6.29 cm
2. B. 7.43 cm
3. C. 8.73 cm (Correct answer)
4. D. 11.41 cm

Correct Answer

Option C — 8.73 cm

Detailed Solution & Explanation

The correct answer is 8.73 cm.

Key Points

• A fully immersed solid displaces oil equal to its own volume, so the rise in level measures that volume.
• Volume of the hemisphere = (2/3)πr³ = (2/3)π(7)³ = (2/3)π(343) ≈ 718.4 cm³.
• The displaced oil forms a cylinder of the container's radius R and height equal to the 3 cm rise:
  • πR² × 3 = 718.4
• So R² = 718.4 / (3π) ≈ 76.2, giving R ≈ 8.73 cm.
• Use (2/3)πr³ for a hemisphere, not (4/3)πr³ — taking the full sphere doubles the volume and lands on option D.

Additional Information

• Volume displacement is the standard method for irregular or immersed solids: the volume of the object equals the volume of liquid displaced.
• The relevant formulae: sphere = (4/3)πr³, hemisphere = (2/3)πr³, cylinder = πr²h and cone = (1/3)πr²h.
• Set displaced volume equal to πR²h, where h is the rise in level, and solve for the unknown — the same equation whether the container widens or narrows.

प्रश्न (हिन्दी में)

एक ऊर्ध्वाधर बेलनाकार बर्तन तेल से भरा है। 7 सेमी त्रिज्या वाला एक ठोस अर्धगोलाकार पत्थर पूरी तरह से उसमें डूब जाता है, और तेल का स्तर 3 सेमी बढ़ जाता है। बेलन की त्रिज्या क्या है?

1. A. 6.29 सेमी
2. B. 7.43 सेमी
3. C. 8.73 सेमी
4. D. 11.41 सेमी