Area & Volume, General Mathematics

Question: Three spherical metal balls of radii `6 cm, 8 cm` and R cm are melted into a solid sphere of radius `12 cm.` The value of R is :
A
`8 cm`

B
`10 cm`

C
`14 cm`

D
`18 cm`

Note: `4/3 π xx (6)^3 + 4/3 π xx (8)^3 + 4/3 π xx R^3` `= 4/3 π xx (12)^3.` `:. 4/3 π xx [216 + 512 + R^3]` `= 4/3 π xx 1728` `or 728 + R^3 = 1728` `or R^3 = 1000.` So, R = 10 cm.`
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Question: If the volume of a sphere is divided by its surface area, the result is `27 cm.` The radius of the sphere is :
A
`81 cm`

B
``9 cm`

C
`54 cm`

D
`36 cm`

Note: `(4/3 π R^3)/(4 π R^3) = 27 ⇒ R` `= 81 cm.`
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Question: A cone and a sphere have equal radii and equal volumes. The ratio of the diameter of the sphere to the height of the cone is :
A
`3 : 1`

B
`1 : 3 `

C
`6 : 1`

D
`1 : 2`

Note: Let radius of each be R and height of the cone be H. Then ,`4/3 π R^3` `= 1/3 π R^3 H or R/H` `= 1/4 or (2R/H)` `= 2/4 = 1/2.`
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Question: The radius of a sphere is increased by `50%`. The increase in the surface area of the sphere is :
A
`100%`

B
`125%`

C
`150%`

D
None

Note: Let original radius = R. Then, original volume `= 4/3 π R^3 = v (say)` New volume `= 150/100 R` `= 3/2 R.` New volume `= 4/3 π (3/2 R)^3` `= (4/3 π R^3). 27/8` `27/ 8 V.` Increase % `= (19/8 V xx 1/V xxx 100)%` `= 237.5%.`
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Question: How many bullets can be made out of a cube of lead whose edge measures `22 cm`, each bullet being `2 cm` in diameter ?
A
`5324`

B
`2662`

C
`1347`

D
`2541`

Note: Number of bullets`= (Volume of the cube)/(Volume of bullet)` `= ((22 xx 22 xx 22)/(4/3 xx 22/7 xx 1 xx 1 xx 1))` `= 2541.`
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Question: A spherical ball of lead, `3` cm in diameter is melted and recast into three spherical balls. The diameter of two of these are `1.5` cm and `2` cm respectively. The diameter of the third ball is :
A
`2.66 cm`

B
`2.5 cm`

C
`3 cm`

D
`3.5 cm`

Note: `4/3 π xx (3/4)^3 + 4/3 π xx (1)^3 + 4/3 π xx x^3` `= 4/3 π xx (3/2)^3` `or 27/64 + 1 + x^3` `= 27/8` `or x^3 = 125/64` `= (5/4)^3` `or x = 5/4.` :. Radius of the third ball is `5/4 cm`. Hence, the diameter of this ball is `5/2 cm = 2.5 cm.`
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Question: If the radius of a sphere is doubled, then its surface area is increased by :
A
`50%`

B
`100%`

C
`200%`

D
`300%`

Note: Let original radius be R. Then, original area`= 4π R^2.` New radius`= 2R,` New area`= 4π (2R)^2` `= 16 π R^2.` Increase %`= ((12π R^2)/(4π R^2) xx 100)%` `= 300%`.
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Question: If a solid sphere of radius `10 cm` is moulded into `8` spherical solid balls of equal radius , then the radius of each such ball is :
A
`1.25 cm`

B
`2.5 cm`

C
`3.75 cm`

D
`5 cm`

Note: VOlume of each ball `= 1/8 xx (4/3 π xx 10 xx 10 xx 10) cm^3. :. 4/3 π R^3` `= 1/8 xx 4/3 π xx 10 xx 10 xx 10` `or R^3 = (10/2)^3 = 5^3.` `So, R = 5.`
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Question: How many lead shorts each `0.3` cm in diameter can be made from a cuboid of dimensions `9 cm xx 11 cm xx 12 cm ?`
A
`7200`

B
`8400`

C
`84000`

D
`72000`

Note: Volume of each lead shot `= 4/3 π xx (0.3/2)^3` `= 4/3 xx 22/7 xx 27/8000` `= 99/7000 cu. cm.` :. Number of lead shots `= (9 xx 11 xx 12 xx 7000/99)` `= 84000.`
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Question: A metallic sphere of radius `10.5 cm is melted and recast into small cones, each of radius `3.5` cm and height `3` cm. The number of such cones will be :
A
`21`

B
`63`

C
`126`

D
`130`

Note: Volume of sphere `= (4/3 π xx 21/2 xx 21/2 xx 21/2) cm^3` `= (3087 π)/2 cm^3.` VOlume of each cone `= (1/3 π xx 7/2 xx 7/2 xx 3) cm^3` `= (49 π)/4 = cm^3.` :. Number of `2` cones `= ((3087 π)/2 xx 4/(49 π))` `= 126.`
1. Report

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Question: Three spherical metal balls of radii `6 cm, 8 cm` and R cm are melted into a solid sphere of radius `12 cm.` The value of R is :

Question: If the volume of a sphere is divided by its surface area, the result is `27 cm.` The radius of the sphere is :

Question: A cone and a sphere have equal radii and equal volumes. The ratio of the diameter of the sphere to the height of the cone is :

Question: The radius of a sphere is increased by `50%`. The increase in the surface area of the sphere is :

Question: How many bullets can be made out of a cube of lead whose edge measures `22 cm`, each bullet being `2 cm` in diameter ?

Question: A spherical ball of lead, `3` cm in diameter is melted and recast into three spherical balls. The diameter of two of these are `1.5` cm and `2` cm respectively. The diameter of the third ball is :

Question: If the radius of a sphere is doubled, then its surface area is increased by :

Question: If a solid sphere of radius `10 cm` is moulded into `8` spherical solid balls of equal radius , then the radius of each such ball is :

Question: How many lead shorts each `0.3` cm in diameter can be made from a cuboid of dimensions `9 cm xx 11 cm xx 12 cm ?`

Question: A metallic sphere of radius `10.5 cm is melted and recast into small cones, each of radius `3.5` cm and height `3` cm. The number of such cones will be :