Mensuration
- From each of the four corners of a rectangular sheet of dimensions 25 cm × 20 cm, a square of side 2 cm is cut off and a box is made. The volume of the box is
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Length of box = 25 – 2 × 2 = 21 cm.
Width of box = 20 – 2 × 2 = 16 cm.
Height of box = 2 cm.
∴ Volume of box = (21 × 16 × 2) cu. cm. = 672 cu. cm.Correct Option: B
Length of box = 25 – 2 × 2 = 21 cm.
Width of box = 20 – 2 × 2 = 16 cm.
Height of box = 2 cm.
∴ Volume of box = (21 × 16 × 2) cu. cm. = 672 cu. cm.
- A solid sphere of radius 3 cm is melted to form a hollow right circular cylindrical tube of length 4 cm and external radius 5 cm. The thickness of the tube is
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Volume of solid sphere = 4 π(3)³ 3
= 36π cu. cm.
Volume of the metal of tube = π(R² – r²)h cu. cm.
where R = 5 cm. r = in-radius
∴ π(R² – r²) h = 36π
⇒ (25 – r²) × 4 = 36⇒ 25 – r² = 36 = 9 4
⇒ r² = 25 – 9 = 16
⇒ r = 16 = 4 cm.
7there4; Thickness of tube = 5 – 4 = 1 cm.Correct Option: A
Volume of solid sphere = 4 π(3)³ 3
= 36π cu. cm.
Volume of the metal of tube = π(R² – r²)h cu. cm.
where R = 5 cm. r = in-radius
∴ π(R² – r²) h = 36π
⇒ (25 – r²) × 4 = 36⇒ 25 – r² = 36 = 9 4
⇒ r² = 25 – 9 = 16
⇒ r = 16 = 4 cm.
7there4; Thickness of tube = 5 – 4 = 1 cm.
- Three small lead spheres of radii 3 cm, 4 cm and 5 cm respectively, are melted into a single sphere. The diameter of the new sphere is
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Volume of new single sphere = 4 π (r1³ + r2³ + r3³) 3 = 4 π (3³ + 4³ + 5³) cu.cm. 3 = 4 π (27 + 64 + 125) cu.cm. 3 = 4 π × 216 cu.cm. 3 ∴ 4 πR³ = 4 π × 216 3 3
where R = radius of new sphere
∴ R³ = 216
⇒ R = ³√216 = ³√6 × 6 × 6 = 6 cm
∴ Diameter of new sphere = 2 × 6 = 12Correct Option: D
Volume of new single sphere = 4 π (r1³ + r2³ + r3³) 3 = 4 π (3³ + 4³ + 5³) cu.cm. 3 = 4 π (27 + 64 + 125) cu.cm. 3 = 4 π × 216 cu.cm. 3 ∴ 4 πR³ = 4 π × 216 3 3
where R = radius of new sphere
∴ R³ = 216
⇒ R = ³√216 = ³√6 × 6 × 6 = 6 cm
∴ Diameter of new sphere = 2 × 6 = 12
- On a rainy day, 60 cm of rain is recorded in a region. What is the volume of water collected in an open and empty rectangular water tank that measures 12 m (length) ×10 m (width) and 50 cm (depth)?
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Water stored in tank = 12 × 10 × 15 = 60 cu. metre 100
= Capacity of tank.Correct Option: C
Water stored in tank = 12 × 10 × 15 = 60 cu. metre 100
= Capacity of tank.
- A hemispherical bowl of internal radius 9 cm, contains a liquid. This liquid is to be filled into small cylindrical bottles of diameter 3 cm and height 4 cm. Then the number of bottles necessary to empty the bowl is
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Volume of bowl = 2 πr³ 3 = 2 π × 9 × 9 × 9 3
= 486p cu. cm.
= volume of liquid Volume of 1 bottle = πR³H= π × 3 × 3 × 4 2 2 ∴ Number of bottles = 486π = 54 9π Correct Option: D
Volume of bowl = 2 πr³ 3 = 2 π × 9 × 9 × 9 3
= 486p cu. cm.
= volume of liquid Volume of 1 bottle = πR³H= π × 3 × 3 × 4 2 2 ∴ Number of bottles = 486π = 54 9π
