Mensuration
- The total surface area of a cube and a sphere are equal. What will be the ratio between their volume?
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Total surface area of cube = 6x²
Surface area of sphere = 4πr²
According to question 6x² = 4πr²⇒ r = √ 6x² = x√6 4π 2√x So, Volume of sphere = 4 π × x√6 × x√6 × x√6 = 4 π × 6x² × x√6 3 2√π 2√π 2 × √π 3 8π × √π ∴ Required ratio = x³ × 3 × 8π × √π = √π = √π : √6 4π × 6x³ × √6 √6 Correct Option: B
Total surface area of cube = 6x²
Surface area of sphere = 4πr²
According to question 6x² = 4πr²⇒ r = √ 6x² = x√6 4π 2√x So, Volume of sphere = 4 π × x√6 × x√6 × x√6 = 4 π × 6x² × x√6 3 2√π 2√π 2 × √π 3 8π × √π ∴ Required ratio = x³ × 3 × 8π × √π = √π = √π : √6 4π × 6x³ × √6 √6
- A rectangular paper sheet of dimensions 22 cm × 12 cm is folded in the form of a cylinder along its length. What will be the volume of this cylinder ? (Take π = 22/7 )
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2πr = 22
⇒2 × 22 × r = 22 ∴ r = 7 7 2
Volume of cylinder = πr²h= 22 × 7 × 7 × 12 = 462 cm³ 7 2 2 Correct Option: B
2πr = 22
⇒2 × 22 × r = 22 ∴ r = 7 7 2
Volume of cylinder = πr²h= 22 × 7 × 7 × 12 = 462 cm³ 7 2 2
- The ratio of the volume of a cube to that of a sphere, which will fit exactly inside the cube, is
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As the sPhere fits exactly inside the cube,
the diameter of sphere will be equal to the edge of cube.
Let the edge of cube be x units.∴ Radius of sphere = x 2
Then,Volume of cube Volume of sphere = 4 = 6 Or 6 : π 4 π 
x 
³ π 3 2 Correct Option: B
As the sPhere fits exactly inside the cube,
the diameter of sphere will be equal to the edge of cube.
Let the edge of cube be x units.∴ Radius of sphere = x 2
Then,Volume of cube Volume of sphere = 4 = 6 Or 6 : π 4 π x ³ π 3 2
- The volume of a sphere and a right circular cylinder having the same radius are equal , The ratio of the diameter of the sphere to the height of the cylinder is
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Volume of sphere = 4 πr³ 3
Volume of cylinder = πr²h
As given,⇒ πr²h = 4 πr³ ⇒ h = 4 r 3 3 ⇒ h = 4 ⇒ h = 4 = 2 r 3 2r 3 × 2 3 ⇒ d = 3 where d = 2r h 2 Correct Option: A
Volume of sphere = 4 πr³ 3
Volume of cylinder = πr²h
As given,⇒ πr²h = 4 πr³ ⇒ h = 4 r 3 3 ⇒ h = 4 ⇒ h = 4 = 2 r 3 2r 3 × 2 3 ⇒ d = 3 where d = 2r h 2
- The size of a rectangular piece of paper is 100 cm × 44 cm. A cylinder is formed by rolling the paper along its length. The volume of the cylinder is (Use π = 22/7)
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When the rectangular sheet is rolled along its length, the length of the sheet forms the circumference of the base of cylinder and breadth of sheet forms the height of cylinder. Circumference = 100
2πr = 100⇒ 2 × 22 × r = 100 7 ⇒ r = 700 = 175 cm 44 11
∴ Volume of the cylinder = πr²h= 22 × 175 × 175 × 44 = 24500 = 35000 cm³ 7 11 11 7 Correct Option: C
When the rectangular sheet is rolled along its length, the length of the sheet forms the circumference of the base of cylinder and breadth of sheet forms the height of cylinder. Circumference = 100
2πr = 100⇒ 2 × 22 × r = 100 7 ⇒ r = 700 = 175 cm 44 11
∴ Volume of the cylinder = πr²h= 22 × 175 × 175 × 44 = 24500 = 35000 cm³ 7 11 11 7
