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The ratio of the volumes of a right circular cylinder and a sphere is 3 ∶ 2. If the radius of the sphere is double the radius of the base of the cylinder; find the ratio of the curved surface area of the cylinder and surface area of the sphere? 
  • a)
    2 ∶ 1
  • b)
    3 ∶ 1
  • c)
    1 ∶ 2 
  • d)
    4 ∶ 1
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The ratio of the volumes of a right circular cylinder and a sphere is ...
Let radius of cylinder be r and radius of sphere be R
Volume of a right circular cylinder = πr2h
Volume of a sphere = (4/3)πR3
Given,
R = 2r
Ratio of the volumes of a right circular cylinder and a sphere is 3 : 2

⇒ h = 16r
Curved Surface area of a right circular cylinder = 2πrh
Surface area of the sphere = 4πR2
∴ Ratio of the surface areas of the cylinder and the sphere = = 2 : 1
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Most Upvoted Answer
The ratio of the volumes of a right circular cylinder and a sphere is ...
Understanding the Problem
To solve the problem, we need to find the ratio of the curved surface area of a right circular cylinder to the surface area of a sphere based on the given conditions.
Given Information
- The ratio of the volumes of the cylinder to the sphere is 3:2.
- The radius of the sphere (R) is double the radius of the base of the cylinder (r), so R = 2r.
Formulas to Use
- Volume of a cylinder (V_cylinder) = πr²h
- Volume of a sphere (V_sphere) = (4/3)πR³
- Curved surface area of a cylinder (CSA_cylinder) = 2πrh
- Surface area of a sphere (SA_sphere) = 4πR²
Volume Ratio Calculation
Given the ratio of the volumes:
- 3/2 = (πr²h) / [(4/3)π(2r)³]
- Simplifying gives: 3/2 = (3r²h) / (32r³)
- From this, we can derive h = (16/3)r.
Surface Area Calculation
Now, we calculate the curved surface areas:
- CSA_cylinder = 2πrh = 2πr*(16/3)r = (32/3)πr²
- SA_sphere = 4πR² = 4π(2r)² = 16πr²
Ratio of Areas
To find the ratio of the curved surface areas:
- Ratio = CSA_cylinder / SA_sphere = [(32/3)πr²] / [16πr²]
- Simplifying gives: Ratio = (32/3) / 16 = 32 / 48 = 2 / 3.
Thus, the required ratio of the curved surface area of the cylinder to the surface area of the sphere is 2:1.
Conclusion
Therefore, the answer is option 'A': 2:1.
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The ratio of the volumes of a right circular cylinder and a sphere is 3 ∶ 2. If the radius of the sphere is double the radius of the base of the cylinder; find the ratio of the curved surface area of the cylinder and surface area of the sphere?a)2 ∶ 1b)3 ∶ 1c)1 ∶ 2d)4 ∶ 1Correct answer is option 'A'. Can you explain this answer?
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