Area & Volume, General Mathematics

Question: Two metallic right circular cones having their heights `4.1` cm and `4.3` cm and the radii of their bases `2.1` cm each, have been melted together and recast into a sphere. The diameter of the sphere is :
A
`4.2 cm`

B
`1.4 cm`

C
`3.5 cm`

D
`6.3 cm`

Note: Volume of `2`cones `= [1/3 π xx (2.1)^2 xx 4.1 + 1/3 π xx (2.1)^2 xx 4.3] cm^3.` `= 1/3 π xx (2.1)^2 (8.4) cm^3.` Volume of sphere `= 1/3 π xx (2.1)^2 (8.4) cm^3.` `:. 4/3 π R^3` `= 1/3 π (2.1)^3 .4` `or R = 2.1 cm.` `:. 4/3 π R^3` :. Diameter of the sphere `= 4.2 cm.`
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Question: A sphere of radius `6.3` cm is melted and cast into a right circular cone of height `25.2` cm. The radius of the base of the cone is :
A
`6.3 cm`

B
`2.1 cm`

C
`2 cm`

D
`3 cm`

Note: `4/3 π xx (6.3)^3` `= 1/3 π R^2 xx 25.2 ⇒ R^2` `= (4 xx 6.3 xx 6.3 xx 6.3)/25.2` `= (6.3)^2. `:. R = 6.3 cm.`
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Question: If the height and diameter of a right circular cylinder are `32` cm and `6` cm respectively, then the radius of the sphere whose volume is equal to the volume of the cylinder is :
A
`3 cm`

B
`4 cm`

C
`6 cm`

D
None

Note: `4/3 π R^3` `= π xx 3 xx 3 xx 32 ⇒ R` `= 6 cm.`
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Question: A hemisphere of lead of radius`6` cm is cast into a right circular cone of height `75` cm. The radius of the base of the cone is :
A
`1.4 cm`

B
`2 cm`

C
`2.4 cm`

D
`4.2 cm`

Note: `2/3 π xx 6 xx 6 xx 6` `= 1/3 π xx R^2 xx 75` `or R^2 = (2 xx 6 xx 6 xx 6)/75` `or R = 12/5` `= 2.4 cm.`
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Question: The diameter and the slant height of a conical tomb are `28` m and `50` m respectively. The cost of whitewashing its curved surface at the rate of `80` paise per square metre is :
A
`Rs. 2640`

B
`Rs.1760`

C
`Rs.264`

D
`Rs. 176`

Note: `r = 14` m and `1 = 50 m.` :. Area of curved surface `= π r/ = (22/7 xx 14 xx 50) m^2` `= 2200 m^2.` :. Cost of white washing `= Rs. (2200 xx 80/100)` `= Rs. 1760.`
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Question: The height and the radius of the base of a cone are each increased by `100%`. The volume of the new cone becomes how many times of the volume of the original cone ?
A
`8 times`

B
`6 times`

C
`4 times`

D
`3 times`

Note: Let radius = r & height = h Then, volume`=1/3 π r^2 h.` New radius`= 200/100 r = 2r` and new height `= 200/100 h = 2h.` :. New volume `= 1/3 π xx (2r)^2 xx 2h` `= 8 xx 1/3 π r^2 h` `= 8 xx `(original volume)
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Question: The radius of two hemisphere vessels are `6.4` litres and `21.6` litres. The areas of inner curved surface of the vessels will be in the ratio of :
A
`4 : 9`

B
`2 : 3`

C
`√2 : √3`

D
`16 : 81`

Note: `(2/3 π R^3)/(2/3 πr^3)` `= 6.4/21.6 or (R/r)^3` `= 8/27 = (2/3)^3.` `So, R/r = 2/3.` :. Ratio of curved surface areas `= (2π R^2)/(2π r^2)` `= (R/r)^2` `= 4/9.`
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Question: If a solid sphere of radius r is melted and cast into the sphape of a solid cone of height r, then the radius of the base of the cone is :
A
`2r`

B
`r`

C
`4r`

D
`3r`

Note: `4/3 π r^3 ` `= 1/3 π R^2 xx r ⇒ R^2` `= 4r^2 ⇒ R = 2r.`
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Question: A right cylinder and a right circular cone have the same radius and the same volume.The ratio of the height of the cylinder to that of the cone is :
A
`3 : 5`

B
`2 : 3`

C
`3 : 1`

D
`1 : 3`

Note: Let radius of each be r, the height of cylinder be H & the height of the cone be h. Then, `π r^2 H = 1/3 π r^2 h` `or H/h = 1/3, i. e. 1 : 3.`
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Question: A right cylindrical vessel is full of water. How many right cones having the same radius and height as those of the right cylinder will be needed to store that water ?
A
`2`

B
`3`

C
`4`

D
`8`

Note: Let radius of each be r & height of each be h. Then, volume of cylinder `= π r^2 h.` volume of `1` cone `1/3 π r^2 h.` Number of cones needed `= (π r^2h)/(1/3 π r^2 h)` `= 3.`
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Question: Two metallic right circular cones having their heights `4.1` cm and `4.3` cm and the radii of their bases `2.1` cm each, have been melted together and recast into a sphere. The diameter of the sphere is :

Question: A sphere of radius `6.3` cm is melted and cast into a right circular cone of height `25.2` cm. The radius of the base of the cone is :

Question: If the height and diameter of a right circular cylinder are `32` cm and `6` cm respectively, then the radius of the sphere whose volume is equal to the volume of the cylinder is :

Question: A hemisphere of lead of radius`6` cm is cast into a right circular cone of height `75` cm. The radius of the base of the cone is :

Question: The diameter and the slant height of a conical tomb are `28` m and `50` m respectively. The cost of whitewashing its curved surface at the rate of `80` paise per square metre is :

Question: The height and the radius of the base of a cone are each increased by `100%`. The volume of the new cone becomes how many times of the volume of the original cone ?

Question: The radius of two hemisphere vessels are `6.4` litres and `21.6` litres. The areas of inner curved surface of the vessels will be in the ratio of :

Question: If a solid sphere of radius r is melted and cast into the sphape of a solid cone of height r, then the radius of the base of the cone is :

Question: A right cylinder and a right circular cone have the same radius and the same volume.The ratio of the height of the cylinder to that of the cone is :

Question: A right cylindrical vessel is full of water. How many right cones having the same radius and height as those of the right cylinder will be needed to store that water ?