A test tube consists of a hemisphere and a cylinder of the same radius. The volume of the water required to fill the whole tube is 5159 6 cm 3 , and 4235 6 cm 3 of water is required to fill the tube to a level which is 4 cm below the top of the tube. Find the radius of the tube and the length of its cylindrical part.

Volume of water filled in test tube = 5159 6 cm 3 volume of water filled upto 4 cm = 4235 6 cm 3 Let r let the radius of tube and h be the height of test tube. ∴ 2 3 π r 3 + π r 2 h = 5159 6 πr 2 2 3 r + h = 5159 6 πr 2 3 2 r + 3 h = 5159 6 ⇒ πr 2 ( 2 r + 3 h ) = 5159 6 . . . . . . . . . . . . . ( i ) and 2 3 πr 3 + πr 2 ( h - 4 ) = 4235 6 ⇒ πr 2 2 3 r + h - 4 = 4235 6 ⇒ πr 2 3 2 r + 3 h - 12 = 4235 6 ⇒ πr 2 ( 2 r + 3 h - 12 ) = 4235 2 . . . . . . . ( ii ) Dividing ( i ) and by ( ii ) 2 r + 3 h 2 r + 3 h - 12 = 5159 4235 . . . . . ( iii ) Subtracting ( ii ) from ( i ) πr 2 ( 12 ) = 5159 2 - 4235 2 = 924 2 ⇒ r 2 = 924 × 7 2 × 12 × 22 = 7 × 7 2 × 2 ⇒ r 2 = 49 4 ⇒ r = 7 2 = 3 . 5 cm Substituting value of r in equation ( iii ) 2 × 7 2 + 3 h 2 × 7 2 + 3 h - 12 = 5159 4235 ⇒ 7 + 3 h 7 + 3 h - 12 = 5159 4235 ⇒ 7 + 3 h 7 + 3 h - 12 = 469 385 ⇒ 2695 + 1155 h = 1407 h - 2345 ⇒ 252 h = 5040 ⇒ h = 20

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The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter $10 cm,$ and the other dimensions are as shown.
Calculate:

The density of the material if its total weight is $1.7 kg .$

Important Questions on Cylinder, Cone and Sphere (Surface Area and Volume)

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10th ICSE

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Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 13

A solid, consisting of a right circular cone standing on a hemisphere, is placed upright in a right circular cylinder, full of water, and
touches the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is

3 cm

and its height is

6 cm;

the radius of the hemisphere is

2 cm

and the height of cone is

4 cm .

Give your answer to the nearest cubic centimetre.

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Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 14

A metal container in the form of a cylinder is surmounted by a hemisphere of the same radius. The internal height of the cylinder is

7 m

and the internal radius is

3 - 5 m .

Calculate the total area of the internal surface, excluding the base.

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Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 14

A metal container in the form of a cylinder is surmounted by a hemisphere of the same radius. The internal height of the cylinder is $7 m$ and the intemal radius is $3.5 m$ Calculate:
the internal volume of the container in $m^{3} .$

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Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 15

An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is

85 m

and height of the cylindrical part is

50 m .

If the diameter of the base is

168 m,

find the quantity of canvas required to make the tent. Allow

20 %

extra for fold and for stitching. Give your answer to the nearest

m^{2} .

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10th ICSE

IMPORTANT

Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 16

A test tube consists of a hemisphere and a cylinder of the same radius. The volume of the water required to fill the whole tube is $\frac{5159}{6} {cm}^{3},$ and $\frac{4235}{6} {cm}^{3}$ of water is required to fill the tube to a level which is $4 cm$ below the top of the tube. Find the radius of the tube and the length of its cylindrical part.

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10th ICSE

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Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 17

A solid is in the form of a right circular cone mounted on a hemisphere. The diameter of the base of the cone, which exactly coincides with
hemisphere, is

7 cm

and its height is

8 cm .

The solid is placed in a cylindrical vessel of internal radius

7 cm

and height

10 cm .

How much water, in

{cm}^{3},

will be required to fill the vessel completely

?

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10th ICSE

IMPORTANT

Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 18

Two solid spheres of radii

2 cm

and

4 cm

are melted and recast into a cone of height

8 cm .

Find the radius of the cone so formed.

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10th ICSE

IMPORTANT

Concise Mathematics>Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)>EXERCISE>Q 19

A certain number of metallic cones, each of radius

2 cm

and height

3 cm,

are melted and recast into a solid sphere of radius

6 cm .

Find
the number of cones used.

The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter 10 cm, and the other dimensions are as shown. Calculate: The density of the material if its total weight is 1.7 kg.

Important Questions on Cylinder, Cone and Sphere (Surface Area and Volume)

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The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter $10 cm,$ and the other dimensions are as shown.
Calculate:

The density of the material if its total weight is $1.7 kg .$