Mensuration
- A rectangular block of metal has dimensions 21 cm, 77 cm and 24 cm. The block has been melted into a sphere. The radius of the sphere is (Take = 22/7)
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Volume of block = 21 × 77 × 24 cu. cm.
Let the radius of sphere be r cm, then4 πr³ = 21 × 77 × 24 3 ⇒ r³ = 21 × 77 × 24 × 3 × 7 4 × 22
= 21 × 7 × 3 × 3 × 7
= 3³ × 7³
⇒ r = 3 × 7 = 21 cmCorrect Option: A
Volume of block = 21 × 77 × 24 cu. cm.
Let the radius of sphere be r cm, then4 πr³ = 21 × 77 × 24 3 ⇒ r³ = 21 × 77 × 24 × 3 × 7 4 × 22
= 21 × 7 × 3 × 3 × 7
= 3³ × 7³
⇒ r = 3 × 7 = 21 cm
- The radius of cross-section of a solid cylindrical rod of iron is 50 cm. The cylinder is melted down and formed into 6 solid spherical balls of the same radius as that of the cylinder. The length of the rod (in metres) is
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Let Length of rod = x cm
∴ Volume of cylinder = Volume of 6 spheres⇒ πr²h = 6 × 4 πr³ 3 ⇒ πr²h = 6 × 4 × r = 8 × 50 = 400 cm = 4 metre 3 Correct Option: D
Let Length of rod = x cm
∴ Volume of cylinder = Volume of 6 spheres⇒ πr²h = 6 × 4 πr³ 3 ⇒ πr²h = 6 × 4 × r = 8 × 50 = 400 cm = 4 metre 3
- Two right circular cones of equal height of radii of base 3 cm and 4 cm are melted together and made to a solid sphere of radius 5 cm. The height of a cone is
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Volume of cone = Volume of sphere
⇒ 1 πr1²h + 1 πr2²h = 4 πr³ 3 3 3
⇒(r1² + r2²)h = 4r³
⇒ (9 + 16)h = 4 × 53
⇒ 25h = 4 × 125
⇒ h = 4 × 5 = 20 cmCorrect Option: D
Volume of cone = Volume of sphere
⇒ 1 πr1²h + 1 πr2²h = 4 πr³ 3 3 3
⇒(r1² + r2²)h = 4r³
⇒ (9 + 16)h = 4 × 53
⇒ 25h = 4 × 125
⇒ h = 4 × 5 = 20 cm
- A tank 40 m long, 30 m broad and 12 m deep is dug in a field 1000 m long and 30 m wide. By how much will the level of the field rise if the earth dug out of the tank is evenly spread over the field ?
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Volume of earth taken out = 40 × 30 × 12 = 14400 cu.metre
Area of the rectangular field = 1000 × 30 = 30000 sq. metre
Area of the region of tank = 40 × 30 = 1200 sq. metre
Remaining area = 30000 – 1200 = 28800 sq. metreIncrease in level = 14400 = 0.5 metre 28800 Correct Option: C
Volume of earth taken out = 40 × 30 × 12 = 14400 cu.metre
Area of the rectangular field = 1000 × 30 = 30000 sq. metre
Area of the region of tank = 40 × 30 = 1200 sq. metre
Remaining area = 30000 – 1200 = 28800 sq. metreIncrease in level = 14400 = 0.5 metre 28800
- A right pyramid 6 m hight has a square base of which the diagonal is √1152 m. Volume of the pyramid is
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Area of the base of pyramid = 1 × (diagonal)² 2 = 1 × 1152 = 576 sq. metre 2 Volume of pyramid = 1 × Area of base × Height 3 Volume of pyramid = 1 × 576 × 6 = 1152 cu. mtere 3 Correct Option: D
Area of the base of pyramid = 1 × (diagonal)² 2 = 1 × 1152 = 576 sq. metre 2 Volume of pyramid = 1 × Area of base × Height 3 Volume of pyramid = 1 × 576 × 6 = 1152 cu. mtere 3
