M!:NS uR1i710N' SOLIDS Note : Until and unless mentioned, take
22
1t
7
1. The height of a right circular cone is 24 cm and the radius of its base is 7 cm.
Calculate: (i) the slant height of the cone;
(ii) the lateral surface area of the cone;
(iii) the total surface area of the cone;
(iv) the volume of the cone.
2. The height of a right circular cone is 8 cm and the diameter of its base is 12 cm.
Calculate : (i) the slant height of the cone;
(ii) the total surface area of the cone;
(iii) the volume of the cone.
(Take n = 3·14)
3. Find the volume and the total surface area of a cone having slant height 17 cm and radius
of the base 15 cm (Take n = 3·14).
4. The height of a cone is 7 cm and the diameter of its base is 7 cm. Calculate the total surface
area of the cone, nearest to two places of decimal.
5. The volume of a right circular cone is 660 cm 3 and the diameter of its base is 12
Calculate : (i) the height of the cone;
(ii) the slant height of the cone;
(iii) the total surface area of the cone.
6. The total surface area of a right circular cone of slant height 20 cm is
Calculate: (i) its radius in cm;
(ii) its volume in cm 3 , in terms of n.
7. The radius and the height of a right circular cone are in the ratio of 5 : 12 and its
is 2512 cm 3 . Find the curved surface area and the total surface area of the
(Take n = 3'14).
8. The circumference of the base of a 12 m high conical tent is 66 m. Find the volume of
contained in it.
9. The curved surface area of a cone is 4070 cm2 and its diameter is 70 cm. Calculate ..
(i) slant height (m height (iii) volume.
10. How many metres of canvas 1·25 m wide will be needed to make a conical tent whose
radius is 17'5 m and height 6 m?
11. A circus tent is cylindrical to a height of 3 metres and conical above it. If its diameter
105 m and the slant height of the conical portion is 53 m, calculate the length of the
2·5 m wide to make the required tent.
[Hint. Area of canvas
22 105
22 105 x53 ) m 2
= ( 2x-x-x3+-x
7
2
7
2
.
Length of canvas
= (:;~) m.J
~J.•. . A solid metallic cylinder of base radius 3 em and height 5 cm is melted to form cones, each
of height 1 cm and base radius 1 mm. Find the number of cones.
Volume of cylinder ]
.
.
Volume of 1 cone
~conical vessel, whose internal radius is 12 cm and height 50 cm, is full of liquid. The
contents are emptied into a cylindrical vessel with internal radius 10 cm. Find the height
to which the liquid rises in the cylindrical vessel.
[Hint. Find the height of the cylinder with radius 10 cm and volume equal to the volume
of the conical vessel.]
[Hint. N urnber 0 f cones =
/cr
1. The surface area of a solid metallic sphere is 1256 sq.cm. It is melted and recast into a solid right
circular cone of radius 2.5 cm and height 8 cm. Calculate:
(i) the radius of solid sphere, and
(ii) the number of cones recast. (2000)
2. An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the te nt abofiv~ !~e
ground is 85 m and height of the cylindrical part is 50 m. If the diameter of the base is 168 m, In
e
quantity of canvas required to make the tent. Allow 20% extra for folds and stitching. Give your answer to
the nearest sq m. (2001)
3. A hollow sphere of internal and external diameter 4 cm and 8 cm respectively is melted into a ~;;o~r
base diameter 8 cm. Find the height of the cone.
4. A vessel is in the form of an inverted cone. Its height is 11 cm and radius of its top which is open,
2.5 cm. It is filled with water upto the rim. When lead shots each of which is a sphere of radius 0.25
are dropped into the vessel, 2/5 of the water overflows out. Find the number of lead shots dropped
(2003)
the vessel.
5. A hemisphere bowl of diameter 7.2 cm is filled completely with chocolate sauce. This sauce is poured'
(2010) .
into an inverted cone of radius 4.8 cm. Find the height of the cone.
6. A solid cone of radius 5 cm and height 8 cm is melted and made into smail spheres of radiUS 0.5 ern.
Find the number of spheres formed.
(2011)
7. A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small
cones of base radius 2 cm and height 8 cm. Find the number of cones.
(2012)
8. A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and
height 8 cm. Calculate the number of cones recast.
(2013)
Solution
Volume of sphere
Number of cones recast = - - - ­
Volume of cone
~1t(15)3
=
~1t(2.5i(8)
A
f)
~')(L)..-U'et$ =270.
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~e\J.l Fundamentals 0/ Mathematics--ICSE
18.18
Note: Use n;
= 2; , unless stated otherwise.
1. Find the volumes and surface areas of the spheres that have radius (r) or diameter (d) as follows:
(i) r 6 cm, (ii) d = 28 cm, (iii) r = 5.75 cm, (iv) d = 5.4 cm.
2. Find the volume and total surface area of a hemisphere of radius 7 cm.
3. The surface area of a sphere is 154 sq cm. Find its radius and volume.
4. The volume of a sphere is 288n; cu cm. Find its radius and surface area.
S. Find the height of a cylinder whose radius is 16.8 cm and the volume same as that of a sphere of
radius 11.2 cm.
6. Assuming the Earth and the Jupiter both to be spheres, find the ratio of the volume of the Jupiter to that
of the Earth if the diameter of the Jupiter is eleven times that of the Earth.
7. How much will the volume and the surface area of a sphere be reduced if the radius of the sphere is halved?
8. If 8 denotes the measure (numerical) of the surface area of a sphere and V denotes the measure of its
volume, show that 8 3 = 36n;V 2 .
9. A piece of butter 3 cm by 3 cm by 12 cm is placed in a hemispherical bowl (or cup) of diameter 7.5 cm. If the butter melts into the cup, will the cup 4cm
overflow? 10. A toy, as shown in Fig 18.31, in the form of a cone is mounted on a hemisphere. The diameter of the base of the cone is 6 cm and its height is 4 cm. Calculate the surface area of the toy. 11. A sphere of radius r has the same volume as that of a cone with a circular Fig. 18.31 base of same radius. Find the height of the cone.
12. How many spherical bullets each of diameter 2 cm can be made out of a cube of lead whose edge
measures 22 cm?
=
t
1
13. The radii of two spheres are in the ratio 1 : 4. Compare their volumes and also their surface areas.
14. A hollow sphere of internal and external diameters 4 cm and 8 cm, respectively is melted into a cone of
base diameter 8 cm. Find the height of the cone.
1S. The volume and diameters of a cone and a sphere are equal. Prove that the height of the cone is twice
the diameter of the sphere.
16. Find the length of the wire of diameter 0.4 cm that can be drawn from a solid sphere of radius 9 cm.
17. A solid metal cylinder of radius 14 cm and height 21 cm is melted down and recast into spheres of
radius 3.5 cm. Calculate the number of spheres that can be made.
18. A buoy is made in the form of a hemisphere surmounted by a right cone whose circular base coincides
with the plane surface of the hemisphere. The radius of the base of the cone is 3.5 rn and its volume is
2/3rds of the hemisphere. Calculate the height of the cone and the surface area of the buoy correct to
two decimal places.
19. Marbles of diameter 1.4 cm are dropped into a beaker containing some water and are fully sub­
merged. The diameter of the beaker is 7 cm. How many marbles have been dropped in it if the water
rises by 5.6 cm?
20. A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top which is open is 2.5 cm.
It is filled with water up to the rim. When lead shots, each of which is a sphere of radius 0.25 cm, are
dropped into the vessel, two-fifths of the water flows out. Find the number of lead shots dropped in the
vessel.
(2003)
21. A cylindrical can whose base is horizontal and of radius 3.5 cm contains sufficient water so that when a
sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can.
Calculate:
(i) the total surface area of the can in contact with water when the sphere is in it, and
(ii) the depth of water in the can before the sphere was put into the can. Take
answers as proper fractions.
1t
to be
~
and give your
Exercise 18.3
1. (i) 905.14 eu em; 452.57 sq em
(iii) 796.65 eu em; 415.64 sq em
2. 718.67 eu em; 462 sq em
5. 6.64 em 7. Seven-eighths, Three-fourths
.: ~.2-
(iu) 82.48 eu em; 91.64 sq em
3.5 em, 179.67 eu em
4. 6 em, 144n sq em
1,331 : 1 No 10. 103.71 sq em
2,541
13. 1: 64; 1 : 16 16. 243 m
17. 72 11. 4r 3.
6.
9.
12.
14. 14 em 18. 4.66 m, 141.16 sq m
19. 150
20. 440 21.
1
(i) 192 2sq em
(ii) 2- em 2:i..
(i) 23 1 m2
(ii) 359-m 3
1
3
22. (i) 175 em3
1
3
(ii) 80 cones. 24. (i) 10 em
f;Y
(iL) 11,498.67 eu em; 2,464 sq em
leSE Examination Questions for Practice 1. (i) 10 em (ii) 80
2. 60,509 m 2
3. 14 em 5. 4.05 em
6. 400 spheres
7. 37 cones
4. 440 (it) 50 em 3