Area & Volume, General Mathematics

Question: The radius of a wife is decreased to one third and its volume remains the same. The new length is how many times the original length ?
A
`1 times`

B
`3 times`

C
`6 times`

D
`9 times`

Note: Let, original radius = r and original length = h. New radius`= (r/3)` and let new length = H. Then, `π r^2 h = π (r/3)^2 xx H or H = 9h.`
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Question: Two cylindrical vessels with radii `15`cm and `10` cm and heights `35` cm ans `15` cm respectively are fiiled with water. If this water is poured into a cylindrical vessel `15` cm in height, then the radius of the vessel is :
A
`17.5 cm`

B
`18 cm`

C
`20 cm`

D
`25 cm`

Note: Volume of new vessel `= [π xx (15)^2 xx 35 + π xx (10)^2 xx 15]` `= 9375 π` `:. πR^2 xx 15` `= 9375π or R^2 = 625.` `So, R = 25 cm.`
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Question: A hollow garden roller `63` cm wide with a girth of `440` cm is made of iron `4` cm thick. The volume of the iron used is :
A
`57636 cm^3`

B
`54982 cm^3`

C
`56372 cm^3`

D
`58752 cm^3`

Note: Circumference of the girth` = 440 cm.` `:. 2πR = 440 ⇒ R` `= (440 xx 1/2xx 7/22)` `= 70 cm.` :. Outer radius` = 70 cm.` Inner radius `= (70 - 4) cm = 66 cm.` Volume of iron `= π [(70)^2 - (66)^2] xx 63` `= (22/7 xx 136 xx 4 xx 63) cm^3` `= 58752 cm^3.`
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Question: If `1` cubic cm of cast iron weighs `21 gms, then the weight of a cast iron pipe of lenght `1` metre `1` metre with a bore of `3` cm and in which the thickness of the metal is `1` cm, is :
A
`1.6 kg`

B
`21 kg`

C
`24.2 kg`

D
`26.4 kg`

Note: Inner radius`= 1.5,` outer radius`= 2.5 cm.` :. Volume of iron `= [π xx (2.5)^2 xx 100 - π xx (1.5)^2 xx 100] cm^3` `= 22/7 xx 100 xx [(2.5)^2 - (1/5)^2] cm^3` `= (8800/7) cm^3` :. Weight of iron `= (8800/7 xx 21/1000) kg` `= 26.4 kg.`
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Question: If the radius of the base of a right circular cylinder is halved, keeping the height same, what is the ratio of the volume of the reduced cylinder to that of the original one ?
A
`1 : 4`

B
`1 : 8`

C
`1 : 2`

D
`8 : 1`

Note: Let original radius = R. Then, new radius` = (R/2).` `(Volume of reduced cylinder)/(Volume of original cylinder)` `=( π xx (R/2) xx h)/(π xx R^2 xx h)` `= 1/4.`
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Question: In what ratio are the volumes of a cylinder, a come and a sphere, if each has the same diameter and the same height ?
A
`1 : 3 : 2`

B
`2 : 3 : 1`

C
`3 : 1 : 2`

D
`3 : 2 : 1`

Note: Let radius = R and height = H. Then, Ratio of their volumes `= πR^2 H : 1/3 πR^2 H : 4/3 π R^3` `= H : 1/3 H : 4/3 R [In sphere H = 2R or R = H/2]` `= H :1/3 H : 4/3 xx H/2 = 3 : 1 : 2.`
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Question: The radius of a sphere is R and the radius of the base as well as the height of a cylinder is R. The ratio of the volume of the sphere to that of the cylinder is :
A
`4 : 3`

B
`3 : 4`

C
`2 : 3`

D
`3 : 2`

Note: `(Volume of sphere)/(Volume of cylinder)` `= (4/3 π R^3)/(πR^2 xx R)` `= 4/3`
1. Report
Question: Two cylinder jars have their diameters in the ratio of `3 : 1` and their heights in the ratio of `1 : 3.` The volumes are in the ratio of :
A
`1 : 2`

B
`3 : 1`

C
`3 : 4`

D
`2 : 3`

Note: Let their radio be `3x` x & heights be y & `3y`. Ratio of their volumes `= (π xx (3x)^2 xx y)/(π xx x^2 xx 3y)` `= 3/1, i e. 3 : 1.`
1. Report
Question: The volume of a sphere is `4851` cu.cm. Its curved surface area is :
A
`1716 cm^2`

B
`1386 cm^2`

C
`1625 cm^2`

D
`3087 cm^2`

Note: `4/3 xx 22/7 xx R^3` `= 4851 ⇒ R^3` `= (4851 xx 3/4 xx 7/22)` `= (21/2)^3.` `So, R = 21/2.` :. curved Surface Area `= (4 xx 22/7 xx 21/2 xx 21/2) cm^2` `= 1386 cm^2.`
1. Report
Question: The curved surface area of a sphere is `5544 sq. cm.` Its volume is :
A
`38808 cm^3`

B
`42304 cm^3`

C
`22176 cm^3`

D
`33951 cm^3`

Note: `4π R^2 = 5544 ⇒ R^2` `= (5544 xx 1/4 xx 1/22)` `= 441.` `:. R = 21.` So, volume `= (4/3 xx 22/7 xx 21 xx 21 xx 21) cm^3` `= 38808 cm^3.
1. Report

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Question: The radius of a wife is decreased to one third and its volume remains the same. The new length is how many times the original length ?

Question: Two cylindrical vessels with radii `15`cm and `10` cm and heights `35` cm ans `15` cm respectively are fiiled with water. If this water is poured into a cylindrical vessel `15` cm in height, then the radius of the vessel is :

Question: A hollow garden roller `63` cm wide with a girth of `440` cm is made of iron `4` cm thick. The volume of the iron used is :

Question: If `1` cubic cm of cast iron weighs `21 gms, then the weight of a cast iron pipe of lenght `1` metre `1` metre with a bore of `3` cm and in which the thickness of the metal is `1` cm, is :

Question: If the radius of the base of a right circular cylinder is halved, keeping the height same, what is the ratio of the volume of the reduced cylinder to that of the original one ?

Question: In what ratio are the volumes of a cylinder, a come and a sphere, if each has the same diameter and the same height ?

Question: The radius of a sphere is R and the radius of the base as well as the height of a cylinder is R. The ratio of the volume of the sphere to that of the cylinder is :

Question: Two cylinder jars have their diameters in the ratio of `3 : 1` and their heights in the ratio of `1 : 3.` The volumes are in the ratio of :

Question: The volume of a sphere is `4851` cu.cm. Its curved surface area is :

Question: The curved surface area of a sphere is `5544 sq. cm.` Its volume is :