Chapter 16 Surface Area and Volume Ex 16.4

Rajasthan Board RBSE Class 10 Maths Chapter 16 Surface Area and Volume Ex 16.4

Question 1.
Find the surface area and the volume of a sphere with radius 1.4 cm.
Solution :
Given
Radius of sphere (r) = 1.4 cm
Surface area = 4πr2
= 4 ×  × 1.4 × 1.4
= 24.64 cm2
Volume of the sphere = πr2
=  ×  × 1.4 × 1.4 × 1.4
= 11.5 cm3
Hence, surface area of sphere = 24.64 cm2 and its volume = 11.5 cm3

Question 2.
The surface area of a sphere is 616 cm2. Find the volume of the sphere.
Solution :
Let radius of sphere be r.
Given, Surface area of sphere = 616 cm2
∴ 4πr2 = 616


Hence, volume of sphere = 1437.33 cm3

Question 3.
Radius of a hemisphere is 4.5 cm. Find its total surface area and volume.
Solution :
Given,
Radius of hemisphere (r) = 4.5 cm
Surface area of hemisphere = 3πr2
Let

Hence, total surface area = 190.93 cm2 and volume = 190.93 cm3

Question 4.
The volume of a sphere is 38808 cm3. Find its surface area.
Solution :
Let radius of the sphere be r,
Volume of sphere = 38808 cm3


Hence, the surface area of sphere = 5544 cm2

Question 5.
A cylinder is made of glass, whose radius and height are 4 cm and 10 cm respectively. By melting it, how many sphere with radii 2 cm each can be recasted?
Solution :
Given, for a cylinder
Radius = 4 cm
And height = 10 cm
Volume = πr2h
= π × 4 × 4 × 10 cm3.
Again the radius of a sphere recasted (r) = 2 cm.
Volume of sphere = πr2
=  × π × (2)3
Let the number of sphere recasted be n, then
Volume of the cylinder = n × volume of sphere

Hence, the number of sphere recasted = 15

Question 6.
The thickness of hollow spherical shell is 2 cm. If its external radius is 8 cm find the volume of metal used in it.
Solution :
Given,
External radius of hollow spherical shell r1 = 8 cm.
Internal radius r2 = 8 – 2 = 6

Hence, volume of the metal used = 1240.38 cm3

Question 7.
How many cones with radius 3 cm and height 6 cm can be formed by melting a metallic sphere with radius 9 cm.
Solution :
Given,
Radius of sphere (r) = 9 cm
Volume of the sphere = πr3
=  ×  × 9 × 9 × 9 cm3
For a cone,
Radius (R) = 3 cm
Height (h) = 6 cm
Volume of the cone = πr2h
=  ×  × 3 × 3 × 6 cm3
Let the number of cones recasted be n.
∴ Volume of sphere = n × volume of a cones

Hence, the number of cones recasted = 54.

Question 8.
Eight spheres of same volumes are recasted by melting a metallic sphere with radius of 10 cm. Find the volume of the recasted sphere.
Solution :
Given,
Radius of the metallic sphere (r) = 10 cm
∴ Volume of sphere = πr3
=  × π × r × (10)3
Let the radius of each recasted spheres be R.
∴ The volume of large sphere =8 × volume of spheres with radius R

Hence, radius of the sphere casted = 5 cm.
Surface area of sphere = 4πR2
= 4 × π × (5)2
=4 × π × 25
= 100π cm2
Hence, surface area of casted sphere = 100π cm2

Question 9.
If surface area of a sphere ¡n 5544 cm2, then find its volume.
Solution :
Let radius of sphere be r, then
Given, surface area of sphere = 5544 cm2
4πr2 = 5544

Hence, volume of sphere = 38808 cm3

Question 10.
The measures of a solid cuboid are 66 cm, 42 cm and 21 cm respectively. How many spheres with diameter 4.2 cm each can be recasted by melting it.
Solution :
Length (l) = 66 cm
Breadth (b) = 42 cm
and height (h) = 21 cm
Volume of cuboid = l × b × h
= 66 × 42 × 21 cm3
Diameter of sphere recasted = 4.2 cm
radius (r) =  = 2.1 cm
Volume of sphere = π × (2.1)3
Let by melting the cuboid n spheres are formed.
∴ Volume of cuboid = n × volume of sphere

Hence, the number of sphere reformed = 1500

Question 11.
A sphere with diameter 6 cm is put into the water filled in a cylindrical vessel with diameter 12 cm. How much the water level in the vessel will rise?
Solution :
Given,
Diameter of sphere = 6 cm
radius (r) =  = 3 cm
Volume of sphere = πr3
=  ×  × (3)3
Diameter of cylindrical vessel = 12 cm
radius (R) =  = 6 cm
When the sphere is put into the vessel, the water level rises by h.
∴ Volume of sphere = volume of water rise into cylinder.

Hence, water level will rise in vessel = 1 cm.

Question 12.
A hemispherical bowl with internal radius 9 cm is filled with liquid This liquid is to fill into the smaller cylindrical bottle with diameter 3 cm and height 4 cm. How many bottle will be needed to fill whole liquid?
Solution :
Given,
Radius of hemispherical bowl (r) = 9 cm
Volume of the bowl =  × π × r3
=  × π × (9)3cm3
Diameter of cylindrical bottle = 3 cm
Radius =  cm
And height = 4 cm
Volume of the bottle = πr2h
= π × 4
Let n bottles will be needed to be filled the whole Liquid
∴ According to the question.
Volume of hemispherical bowl = n × volume of cylindrical bottles.

Hence, 54 bottles will be needed.

Question 13.
The diameter of a sphere is 0.7 cm. From a water tank, 3000 spheres completely filled with water is thrown out. Find the volume of the
water thrown out.
Solution :
Given,
Diameter of sphere = 0.7 cm
Radius (r) = cm
∴ Volume of sphere V = πr3
= πcm3
Volume of water thrown out = 3000 × volume of sphere
= 3000 × π

= 539 cm3

Question 14.
The external and internal diameters of hemispheric bowl are 43 cm and 42 cm respectively. If the coloring cost is paise 7 per cm2, find the total cost of coloring the bowl.
Solution :
Given,
External diameter of hemispherical bowl = 43 cm
∴ External radius (r1) =  cm
Internal diameter = 42 cm
∴ Internal radius (r2) =  = 21 cm
External surface area of bowl = 2πr12
And internal surface area = 2πr22

Hence, cost of coloring the bowl = ₹ 397.43