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A hollow sphere of internal and external diameter 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.
  • a)
    12 cm
  • b)
    16 cm
  • c)
    14 cm
  • d)
    7 cm
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
A hollow sphere of internal and external diameter 4 cm and 8 cm respec...
The volume of cone = Volume of metal in the sphere
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Community Answer
A hollow sphere of internal and external diameter 4 cm and 8 cm respec...
Volume of the Hollow Sphere
To find the height of the cone formed by melting the hollow sphere, we first need to calculate the volume of the hollow sphere.
- External diameter: 8 cm
- Internal diameter: 4 cm
- External radius (R): 8 cm / 2 = 4 cm
- Internal radius (r): 4 cm / 2 = 2 cm
The volume of a hollow sphere is given by the formula:
Volume = (4/3)π(R³ - r³)
Calculating the volume:
- R³ = 4³ = 64
- r³ = 2³ = 8
- R³ - r³ = 64 - 8 = 56
Thus,
Volume of the hollow sphere = (4/3)π(56) = (224/3)π cm³
Volume of the Cone
Now, the volume of the cone formed from this hollow sphere is equal to the volume of the hollow sphere.
- Base diameter of the cone: 8 cm
- Radius of the cone (r_cone): 8 cm / 2 = 4 cm
Let the height of the cone be h. The volume of the cone is given by:
Volume = (1/3)πr_cone²h
Setting the volumes equal:
(224/3)π = (1/3)π(4²)h
Cancelling π and (1/3):
224 = 16h
Solving for h:
h = 224 / 16 = 14 cm
Conclusion
The height of the cone is 14 cm. Thus, the correct answer is option 'C'.
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