Mensuration
- A cylindrical vessel of radius 4 cm. contains water. A solid sphere of radius 3 cm. is dipped into the water until it is completely immersed. The water level in the vessel will rise by
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If the rise in water level be h cm., then
πr²h = 4 πR³ 3
where r = radius of cylindrical vessel,
R = radius of solid sphere⇒ 4² × h = 4 × (3)³ 3 ⇒ h = 4 × 3 × 3 = 9 4 × 4 4
= 2.25 cm.Correct Option: B
If the rise in water level be h cm., then
πr²h = 4 πR³ 3
where r = radius of cylindrical vessel,
R = radius of solid sphere⇒ 4² × h = 4 × (3)³ 3 ⇒ h = 4 × 3 × 3 = 9 4 × 4 4
= 2.25 cm.
- A hollow hemispherical bowl is made of silver with its outer radius 8 cm and inner radius 4 cm respectively. The bowl is melted to form a solid right circular cone of radius 8 cm. The height of the cone formed is
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Volume of the silver used in hollow hemispherical bowl = 2 π(R³ - r³)cu.cm. 3
Where R = external radius r = internal radius= 2 π(8³ - 4³)cu.cm. 3 = 2 π(512 - 64)cu.cm. 3 = 2π × 448 cu.cm. 3 ∴ Volume of cone = 1 πr1²h 3 = 1 π 8² × h 3 ∴ 1 π 8² × h = 2π × 448 3 3 ⇒ h = 2 × 448 = 14 cm. 8 × 8 Correct Option: D
Volume of the silver used in hollow hemispherical bowl = 2 π(R³ - r³)cu.cm. 3
Where R = external radius r = internal radius= 2 π(8³ - 4³)cu.cm. 3 = 2 π(512 - 64)cu.cm. 3 = 2π × 448 cu.cm. 3 ∴ Volume of cone = 1 πr1²h 3 = 1 π 8² × h 3 ∴ 1 π 8² × h = 2π × 448 3 3 ⇒ h = 2 × 448 = 14 cm. 8 × 8
- If the sum of radius and height of a solid cylinder is 20 cm and its total surface area is 880 cm.2 then its volume is
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According to the question, r + h = 20 cm.
Total surface area of cylinder = 2πrh + 2πr² = 2πr (h + r)
∴ 2πr × 20 = 880⇒ πr = 880 = 22 40 ⇒ 22 × r = 22 7 ⇒ r = 22 × 7 = 7 cm. 22
∴ r + h = 20
⇒ h = 20 – 7 = 13 cm.
∴ Volume of cylinder = πr²h= 22 × 7 × 7 × 13 = 2002 cu. cm. 7 Correct Option: C
According to the question, r + h = 20 cm.
Total surface area of cylinder = 2πrh + 2πr² = 2πr (h + r)
∴ 2πr × 20 = 880⇒ πr = 880 = 22 40 ⇒ 22 × r = 22 7 ⇒ r = 22 × 7 = 7 cm. 22
∴ r + h = 20
⇒ h = 20 – 7 = 13 cm.
∴ Volume of cylinder = πr²h= 22 × 7 × 7 × 13 = 2002 cu. cm. 7
- A solid sphere and a solid hemisphere have the same total surface area. The ratio of their volumes is (Take, π= 22/7)
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Radius of solid sphere = R units
Radius of solid hemisphere = r units
According to the question,
4πR² = 3πr²
⇒ 4R² = 3r²⇒ R² = 3 ⇒ R = √3 r² 4 r 2 ∴ Ratio of volumes = 4 πR³ = 2 = 2 = = 3√3 : 4 3 
R 
³ √3 ³ 3√3 2 πr³ r 2 4 3 Correct Option: A
Radius of solid sphere = R units
Radius of solid hemisphere = r units
According to the question,
4πR² = 3πr²
⇒ 4R² = 3r²⇒ R² = 3 ⇒ R = √3 r² 4 r 2 ∴ Ratio of volumes = 4 πR³ = 2 = 2 = = 3√3 : 4 3 R ³ √3 ³ 3√3 2 πr³ r 2 4 3
- The base of a right prism is a trapezium whose lengths of parallel sides are 25 cm. and 11 cm. and the perpendicular distance between the parallel sides is 16 cm. If the height of the prism is 10 cm., then the volume of the prismis
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Area of the base of prism = 1 (sum of parallel sides) × per-pendicular distance 2 = 1 (25 + 11) × 16 2 = 1 × 36 × 16 = 288 sq. cm. 2
∴ Volume of prism = Area of base × height = 288 × 10 = 2880 cu. cm.Correct Option: C
Area of the base of prism = 1 (sum of parallel sides) × per-pendicular distance 2 = 1 (25 + 11) × 16 2 = 1 × 36 × 16 = 288 sq. cm. 2
∴ Volume of prism = Area of base × height = 288 × 10 = 2880 cu. cm.
