RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS

Other Exercises

Question 1.
The radii of the bases of a cylinder and a cone are in the ratio 3 : 4 and their heights are in the ratio 2 : 3. What is the ratio of their volumes ?
Solution:
Radii of the bases of a cylinder and a cone = 3:4
and ratio in their heights = 2:3
Let r1, r2 be the radii and h1 and h2 be their heights
heights of the cylinder and cone respectively,
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 1

Question 2.
If the heights of two right circular cones are in the ratio 1 : 2 and the perimeters of their bases are in the ratio 3 : 4. What is the ratio of their volumes ?
Solution:
Ratio in the heights of two cones =1:2 and ratio in the perimeter of their bases = 3:4
Let r1, r2 be the radii of two cones and ht and h2 be their heights
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 2

Question 3.
If a cone and sphere have equal radii and equal volumes what is the ratio of the diameter of the sphere to the height of the cone ?
Solution:
Let r be the radius of a cone, then
radius of sphere = r
Let h be the height of cone
Now volume of cone = volume of sphere
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 3

Question 4.
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. What is the ratio of their volumes?
Solution:
Let r and h be the radius and heights of a cone, a hemisphere and a cylinder
∴ Volume of cone =  \((\frac { 1 }{ 3 } )\) πr²h
Volume of hemisphere = \((\frac { 2 }{ 3 } )\) πr³
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 4

Question 5.
The radii of two cylinders are in the ratio 3 : 5 and their heights are in the ratio 2 : 3. What is the ratio of their curved surface areas ?
Solution:
Radii of two cylinders are in the ratio = 3:5
and ratio in their heights = 2:3
Let r1, r2 be the radii and h1, h2 be the heights of the two cylinders respectively, then
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 5

Question 6.
Two cubes have their volumes in the ratio 1 : 27. What is the ratio of their surface areas ?
Solution:
Ratio in the volumes of two cubes = 1 : 27
Let a1 and a2 be the sides of the two cubes respectively then volume of the first area = a1³
and volume of second cube = a
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 6

Question 7.
Two right circular cylinders of equal volumes have their heights in the ratio 1 : 2. What is the ratio of their radii ?
Solution:
Ratio the heights of two right circular cylinders = 1:2
Let r1,r2 be their radii and h1, hbe their
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 7

Question 8.
If the volumes of two cones are in the ratio 1 : 4, and their diameters are in the ratio 4 : 5, then write the ratio of their weights.
Solution:
Volumes of two cones are in the ratio =1:4 and their diameter are in the ratio = 4:5
Let r1 and r2 be the radii and h,h2 be their
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 8

Question 9.
A sphere and a cube have equal surface areas. What is the ratio of the volume of the sphere to that of the cube ?
Solution:
Surface areas of a sphere and a cube are equal
Let r be the radius of sphere and a be the edge of cube,
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 9

Question 10.
What is the ratio of the volume of a cube to that of a sphere which will fit inside it?
Solution:
A sphere is fit inside the cube
Side of a cube = diameter of sphere
Let a be the side of cube and r be the radius of the sphere, then
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 10

Question 11.
What is the ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height ?
Solution:
Diameters (or radii), and heights of a cylinder a cone and a sphere are equal,
Let r and h be the radius and height be the cone cylinder, cone and sphere respectively, thus their volumes will be
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 11

Question 12.
A sphere of maximum volume is cut-out from a solid hemisphere of radius r. What is the ratio of the volume of the hemisphere to that of the cut-out sphere?
Solution:
r is the radius of a hemisphere, then
the diameter of the sphere which is cut out of the hemisphere will be r
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 12

Question 13.
A metallic hemisphere is melted and recast in the shape of a cone with the same base radius R as that of the hemisphere. If H is the height of the cone, then write the value of \((\frac { H }{ R } )\).
Solution:
R is the radius of a hemisphere 2
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 13

Question 14.
A right circular cone and a right circular cylinder have equal base and equal height. If the radius of the base and height are in the ratio 5 : 12, write the ratio of the total surface area of the cylinder to that of the cone.
Solution:
Radius and height of a cone and a cylinder be r and h respectively
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 14

Question 15.
A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes ?
Solution:
Let r and h be the radii and heights of the cylinder cone and hemisphere respectively, then
Volume of cylinder = πr²h
Volume of cone = \((\frac { 1 }{ 3 } )\) πr²h
Volume of hemisphere = \((\frac { 2 }{ 3 } )\) πr³
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 15

Question 16.
The radii of two cones are in the ratio 2 : 1 and their volumes are equal. What is the ratio of their heights ?
Solution:
Radii of two cones are in the ratio = 2:1
Let r1, r2 be the radii of two cones and h1, h2 be their heights respectively,
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 16

Question 17.
Two cones have their heights in the ratio 1 : 3 and radii 3:1. What is the ratio of their volumes ?
Solution:
Ratio in heights of two cones = 1:3
and ratio in their ratio = 3:1
Let r1, r2 be their radii and h1, h2 be their
heights, then
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 17

Question 18.
A hemisphere and a cone have equal bases. If their heights are also equal, then what is the ratio of their curved surfaces ?
Solution:
Bases of a hemisphere and a cone are equal
and their heights are also equal
Let r and h be their radii and heights
respectively
∴ r = h1
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 18

Question 19.
If r1 and r2 denote the radii of the circular bases of the frustum of a cone such that r1 > r2 then write the ratio of the height of the cone of which the frustum is a part to the height of the frustum.
Solution:
r1 , r2 are the radii of the bases of a frustum and r1 > r2
Let h1 be the height of cone and h2 be the height of smaller cone
∴ Height of frustum = h1 – h2
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 19

Question 20.
If the slant height of the frustum of a cone is 6 cm and the perimeters of its circular bases are 24 cm and 12 cm respectively. What is the curved surface area of the frustum ?
Solution:
Slant height of a frustum (l) = 6 cm
Perimeter of upper base (P1) = 24 cm
and perimeter of lower base (P2) = 12 cm
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 20

Question 21.
If the areas of circular bases of a frustum of a cone are 4 cm² and 9 cm² respectively and the height of the frustum is 12 cm. What is the volume of the frustum ?
Solution:
In a frustum,
Area of upper base (A1) = 4 cm²
and area of lower base (A2) = 9 cm²
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 21

Question 22.
The surface area of a sphere is 616 cm². Find its radius.
Solution:
Surface area of a sphere = 616 cm²
Let r be the radius, then
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 22

Question 23.
A cylinder and a cone are of the same base radius and of same height. Find the ratio of the value of the cylinder to that of the cone. [CBSE 2009]
Solution:
Let r be the radius of the base of the cylinder
small as of cone
and let height of the cylinder = h
Then height of cone = h
∴ Volume of cylinder =  πr²h
and volume of cone = \((\frac { 1 }{ 3 } )\)  πr²h
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 23

Question 24.
The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, write the height of the frustum. [CBSE 2010]
Solution:
Slant height of frustum (l) = 5 cm
Difference between the upper and lower radii = 4 cm
Let h be height and upper radius r1 and lower radius = r2
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 24

Question 25.
Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?
Solution:
Volume of hemisphere = Surface area of hemisphere (given)
RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes VSAQS 25

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