Area & Volume, General Mathematics

Question: The ratio of total surface area to lateral surface area of a cylinder whose radius is `20` cm. and height of the wire is :
A
`2 : 1`

B
`3 : 2`

C
`4 : 3`

D
`5 : 3`

Note: `(Total surface Area)/(Lateral surface Area)` `= (2πrh + 2πr^2)/(2πrh)` `= (h + r)/h` `= 80/60` `= 4/3`
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Question: The radii of two cylinders are in the ratio of `2 : 3` and their heights are in the ratio of `5 : 3.` The ratio of their volume is :
A
`4 : 9`

B
`9 : 4`

C
`20 : 27`

D
`27 : 20`

Note: Let their radii be `2x, 3x` and heights be `5y, 3y.` Ratio of their volumes `= (π(2x)^2 xx 5y)/(π (3x)^2 xx 3y)` `= 20/27`
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Question: The ratio between the radius of the base and the height of a cylinder is`2 : 3.` If its volume is `12936` cu, the total surface area of cylinder is :
A
`3080 cm^2`

B
`38808 cm^2`

C
`25872 cm^2`

D
`2587.2 cm^2`

Note: Let radius`= 2x & height`= 3x. Then, `22/7 xx (2x)^2 xx 3x` `= 12936 or x^3` `= (12936 xx 7/22 xx 1/12)` `= 343 = 7^3` `:. x = 7.` So,ratius `= 14 cm` & height = 21 cm. :. Total surface area `= (2 xx 22/7 xx 14 xx 21 + 2 xx 22/7 xx 14 xx 14)sq.` cm.` `= 3080 cm^2.`
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Question: A solid cylinder has a total surface area of `231` sq. cm. If its curved surface area is two third of the total surface area, the volume of the cylinder is :
A
`269.5 cm^3`

B
`385 cm^3`

C
`308 cm^3`

D
`363.4 cm^3`

Note: Curved Surface Area `= (2/3 xx 231) cm^2 = 154 cm^2.` `:. 2πrh + 2πr^2` `= 231 ⇒ 154 + 2π r^2` `= 231` `:. r^2 = (77 xx 1/2 xx 7/22)` `= 49/4 or r = 7/2.` Now,`2πrh` `= 154 ⇒ 2 xx 22/7 xx 7/2 xx h` `= 154 or h = 7.` :. Volume `= πr^2h` `= (22/7 xx 7/2 xx 7/2 xx 7) cm^2` `= 269.5 cm^3.`
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Question: The sum of the radius of the base and the height of a solid cylinder is `37` metres. If the total surface area of the cylinder be `1628 sq.`metres, its volume is :
A
`5240 m^3`

B
`4620 m^3`

C
`3180 m^3`

D
None of these

Note: `(h + r)` `= 37` and `2πr (h + r)` `= 1628.` `:. 2πr xx 37 = 1628` or` r = (1628/(2 xx 37) xx 7/22)` `= 7.` `:. r = 7` m and `h = 30 m.` :. Volume`= πr^2h` `= (22/7 xx 7 xx 7 xx 30) m^3` `= 4620 m^3.`
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Question: The ratio between the curved surface area and the total surface area of a right circular cylinder is `1 : 2`.If the total surface area is `616 sq. cm, the volume of the cylinder is ;
A
`1848 cm^3`

B
`1232 cm^3`

C
`1078 cm^3`

D
None of these

Note: `(2πrh)/(2πrh + 2πr^2)` `= 1/2 ⇒ (2πrh)/(2πr (h + r)` `= 1/2 or h/(h + r)` `= 1/2 or h = r.` now, Total surface area `= 2πrh + 2πr^2` `= 4πr^2 [:. h = r]` `:. 4πr^2 616 or r^2.` `= (616 xx 1/4 xx 7/22)` `= 49 or r = 7.` Thus, `h = 7 cm` and `r = 7 cm.` :. Volume `= (22/7 xx 7 xx 7 xx 7) cm^3.` `= 1078 cm.^3`
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Question: A cylindrical vessel `60` cm in diameter is partially water. A sphere `30` cm is diameter into it. The increase in the level of water in the vessel is :
A
`2 cm`

B
`3 cm`

C
`4 cm`

D
`5 cm`

Note: Let h and H be the heights of water level before and after dropping the sphere. Then, `[π xx (30)^2 xx H] - [π xx (30)^2 xx h]` `= 4/3 π xx (15^3` `900 π (H -h)` `= 4500 π or (H - h)` `= 5 cm.`
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Question: A cylindrical vessel of radius `8` cm contains water. A solid sphere of radius `6` cm is lowered into the water until it is completely immersed. The water level in the vessel will rise by :
A
`3.5 cm`

B
`4.5 cm`

C
`4 cm`

D
`7 cm`

Note: `π xx (8)^2 xx h` `= 4/3 π xx (6)^3 or h` `= (4 xx 72)/64` `= 4.5 cm.`
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Question: The curved surface area of a cylindrical pillar is `528` sq. m and its volume is `2772` cu.m. The height of the pillar is :
A
`10.5 m`

B
`7.5 m`

C
`8 m`

D
`5.25 m`

Note: `2 π r h = 528 and π r^2 h = 2772.` `:. (πr^2 h)/(2 π rh)` `= 2772/528 ⇒ r` `= (2 xx 2772)/528` `= 21/2` `:. 2 xx 22/7 xx 21/2 xx h` `= 528 ⇒ h` `= (528/66)` `= 8 m.`
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Question: The number of solid sphere, each of diameter `3` cm that could be moulded to form a solid cylinder of height `54` cm and diameter `4` cm, is :
A
`16`

B
`24`

C
`36`

D
`48`

Note: Volume of cylinder `= π xx 2^2 xx 54` `= (216 π) cm^3` Volume of `1` sphere `= 4/3 π xx (3/2)^3` `= ((9π)/2) cm.^2` Number of spheres `= (216π xx 2/(9π)` `= 48.`
1. Report

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Question: The ratio of total surface area to lateral surface area of a cylinder whose radius is `20` cm. and height of the wire is :

Question: The radii of two cylinders are in the ratio of `2 : 3` and their heights are in the ratio of `5 : 3.` The ratio of their volume is :

Question: The ratio between the radius of the base and the height of a cylinder is`2 : 3.` If its volume is `12936` cu, the total surface area of cylinder is :

Question: A solid cylinder has a total surface area of `231` sq. cm. If its curved surface area is two third of the total surface area, the volume of the cylinder is :

Question: The sum of the radius of the base and the height of a solid cylinder is `37` metres. If the total surface area of the cylinder be `1628 sq.`metres, its volume is :

Question: The ratio between the curved surface area and the total surface area of a right circular cylinder is `1 : 2`.If the total surface area is `616 sq. cm, the volume of the cylinder is ;

Question: A cylindrical vessel `60` cm in diameter is partially water. A sphere `30` cm is diameter into it. The increase in the level of water in the vessel is :

Question: A cylindrical vessel of radius `8` cm contains water. A solid sphere of radius `6` cm is lowered into the water until it is completely immersed. The water level in the vessel will rise by :

Question: The curved surface area of a cylindrical pillar is `528` sq. m and its volume is `2772` cu.m. The height of the pillar is :

Question: The number of solid sphere, each of diameter `3` cm that could be moulded to form a solid cylinder of height `54` cm and diameter `4` cm, is :