Mensuration
- If S1 and S2 be the surface area of a sphere and the curved surface area of the circumscribed cylinder respectively, then S1 is equal to
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S1 = surface area of sphere = 4πr²
S2 = curved surface of the circumscribed cylinder
= 2πRH = 2π (2r) (2r) = 8πr²⇒ 2 πR³= 1 πR³H 3 3 ∴ S1 = 4πr² = 1 S2 8πr² 2 ⇒ S1 = 1 S2 2 Correct Option: B
S1 = surface area of sphere = 4πr²
S2 = curved surface of the circumscribed cylinder
= 2πRH = 2π (2r) (2r) = 8πr²⇒ 2 πR³= 1 πR³H 3 3 ∴ S1 = 4πr² = 1 S2 8πr² 2 ⇒ S1 = 1 S2 2
- A solid metallic sphere of radius 8 cm is melted to form 64 equal small solid spheres. The ratio of the surface area of this sphere to that of a small sphere is
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Volume of the solid metallicsphere = 4 πr³ 3 = 4 × π × (8)³ 3 = 2048 π cm³ 3
Let the radius of the each small sphere be x cm∴ 64 × 4 πx³ = 2048 π 3 3 ⇒ x³ = 2048 = 8 64 × 4
⇒ x = ³√8 = 2cm
∴ Required ratio = 4π. (8)² : 4π (2)² = 64 : 4 = 16 : 1Correct Option: C
Volume of the solid metallicsphere = 4 πr³ 3 = 4 × π × (8)³ 3 = 2048 π cm³ 3
Let the radius of the each small sphere be x cm∴ 64 × 4 πx³ = 2048 π 3 3 ⇒ x³ = 2048 = 8 64 × 4
⇒ x = ³√8 = 2cm
∴ Required ratio = 4π. (8)² : 4π (2)² = 64 : 4 = 16 : 1
- If the radii of two spheres are in the ratio 1 : 4, then their surface area are in the ratio :
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Required ratio = 4πr1² = 
r1 
² = 1 ² = 1 4πr2² r2 4 16
or 1 : 16Correct Option: D
Required ratio = 4πr1² = r1 ² = 1 ² = 1 4πr2² r2 4 16
or 1 : 16
- The total surface area of a metallic hemisphere is 1848 cm2. The hemisphere is melted to form a solid right circular cone. If the radius of the base of the cone is the same as the radius of the hemisphere, its height is
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Total curved surface area of hemisphere = 3πr²,
where r = radius of hemisphere.
∴ 3πr² = 1848⇒ 3 × 22 × r² = 1848 7 ⇒ r² = 1848 × 7 = 196 3 × 22
⇒ r = √196 = 14 cm.Volume of hemisphere = 2 × π × 14 × 14 × 14 cm³ 3 = 5488 π cm³ 3
According to the question, Volume of cone = Volume of hemisphere⇒ 1 πr²h = 5488 π cm³ 3 3
⇒ r²h = 5488
→ 14 × 14 × h = 5488⇒ h = 5488 = 28 cm 14 × 14 Correct Option: C
Total curved surface area of hemisphere = 3πr²,
where r = radius of hemisphere.
∴ 3πr² = 1848⇒ 3 × 22 × r² = 1848 7 ⇒ r² = 1848 × 7 = 196 3 × 22
⇒ r = √196 = 14 cm.Volume of hemisphere = 2 × π × 14 × 14 × 14 cm³ 3 = 5488 π cm³ 3
According to the question, Volume of cone = Volume of hemisphere⇒ 1 πr²h = 5488 π cm³ 3 3
⇒ r²h = 5488
→ 14 × 14 × h = 5488⇒ h = 5488 = 28 cm 14 × 14
- The ratio of the surface area of a sphere and the curved surface area of the cylinder circumscribing the sphere is
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Let Height of the cylinder = 2r,
where r = radius of sphere.
Radius of cylinder = r∴ Surface area of sphere Curved surface area of cylinder = 4πr² = 1 : 1 2πr × 2r Correct Option: B
Let Height of the cylinder = 2r,
where r = radius of sphere.
Radius of cylinder = r∴ Surface area of sphere Curved surface area of cylinder = 4πr² = 1 : 1 2πr × 2r
