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Spheres

Theory and Formula



Question


Q1. The diameter of a metallic sphere is 6cm . It is melted and drawn into a wire having diameter of the cross-section as 0.6 cm. Find the length of the wire .

Solution.
let length of the wire be h cm, since metallic sphere is converted into a cylindrical shaped wire of length h cm .

volume of wire = volume of sphere

π(0.3)2 x h =
4/3
π(3)3

3(0.09)h = 4(27)

0.27h = 4 x 27 ⇒h = 400cm = 4 meters
Solution

Q2. Spheres of radii 6cm , 8cm and 10 cm respectively , are melted to form a single solid sphere . Find the radius of single sphere .

Solution.
4/3
πr3 =
4/3
π(6)3 +
4/3
π(8)3 +
4/3
π(10)3

r = 12 cm

Solution

Q3. A cone and a sphere have equal radii and equal volume . what is the ratio of the radius of the sphere to the height of the cone .

Solution.
1/3
πr12 h =
4/3
πr23

r1 = r2

h = 4r

r/h
=
1/4
⇛ r:h = 1:4
Solution

Q4. The surface area of two spheres are in the ratio 4:9, find the ratio of their volumes.

(a) 5:11 (b) 4:9 (c) 1:2 (d) 8:27

Solution.
(d) 8:27
Solution

Q5. How many spheres of radius 0.5 cm can be created on melting the sphere of radius 4 cm?

(a) 120 (b) 300 (c) 512 (d) 600

Solution.
(c) 512
Solution

Q6. The ratio of volume of the sphere is 27:64 , find the ratio of their radii

Solution.
3:4
Solution

Q7. The surface area of sphere is 144π m2

Solution.
Let the radius of the sphere is r cm

4πr2 = 144 π

⇒ r2 = 36 ⇒ r = 6 cm

∴volume of the sphere =
4/3
πr 3 =
4/3
π(6)3

= 288π cm3
Solution

Q8. The volume of two spheres are in the ratio 125:27, the ratio of their surface area is.

(a) 20:25 (b) 9:25 (c) 25:3 (d) 25:9

Solution.
4/3 π r31 / 4/3 π r32
=
125 / 27


r31 / r32
=
125 / 27


r1 / r2
=
5 / 3


Surface area of sphere = 4π r2

4π r21 / 4π r22
=
r21 / r22
=
(5)2 / (3)2
=
25 / 9


25:9
Solution

Q9. The maximum volume of a cone that can be curved out of a solid hemisphere of radius r, is

(a) πr3/3 (b) πr3 (c) r3/3 (d) πr2/2

Solution.
V = (1/3) π r2h

V = ( 1/3 )π r2 x r

= (1/3) π r3
Solution

Q10. A cone and a hemisphere have equal bases. If their height are also equal then what is the ratio of their curved surface ?

(a) √2 : 3 (b) 1: √2 (c) 1 : 1 (d) 1 : 3

Solution.
(b) 1: √2
Solution