RD Sharma Solutions for Class 10 Chapter 16 Surface Area and Volumes Exercise 16.1

Chapter 16 Surface Areas and Volumes RD Sharma Solutions Exercise 16.1 Class 10 Maths

Chapter Name	RD Sharma Chapter 16 Surface Area and Volumes Exercise 16.1
Book Name	RD Sharma Mathematics for Class 10
Other Exercises	Exercise 16.2 Exercise 16.3
Related Study	NCERT Solutions for Class 10 Maths

Exercise 16.1 Solutions

1. How many balls each or radius 1 cm can be made from a solid sphere of lead of radius 8 cm?

Solution

2. How many spherical bullets each of 5cm in diameter can be cast from a rectangular block of metal 11dm × 1m × 5dm ? 

Solution

Given that a metallic block which is rectangular of diameter 11dm × 1m × 5dm 
Given that diameter of each bullet is 5cm 

3. A spherical ball of radius 3cm is melted and recast into three spherical balls. The radii of the two of balls are 1.9 cm and 2cm. Determine the diameter of the third ball ? 

Solution

Diameter (d) = 2r = 2×2.5 = 5 cm 

∴ Diameter of third ball = 5cm. 

4. 2.2 cubic dm of grass is to be drawn into a cylinder wire 0.25cm in diameter. Find the length of wire ? 

Solution

5. What length of a solid cylinder 2cm in diameter must be taken to recast into a hollow cylinder of length 16cm, external diameter 20cm and thickness 2.5mm ?

Solution

Given that diameter of solid cylinder = 2cm 

Given that solid cylinder is recast to hollow cylinder 

Length of hollow cylinder = 16 cm

External diameter = 20 cm 

Thickness = 2.5 mm = 0.25cm 

Volume of solid cylinder = πr²h
Radius of cylinder = 1cm
so volume of solid cylinder = π(1)²h

Let length of solid cylinder be h 

6. A cylindrical vessel having diameter equal to its height is full of water which is poured into two identical cylindrical vessels with diameter 42cm and height 21cm which are filled completely. Find the diameter of cylindrical vessel ?

Solution

Given that diameter is equal to height of a cylinder 

So h = 2r 
Volume of cylinder = πr²h

7. 50 circular plates each of diameter 14 cm and thickness 0.5cm are placed one above other to form a right circular cylinder. Find its total surface area ? 

Solution

Given that 50 circular plates each with diameter = 14cm 

Radius of circular plates (r) = 7cm 

Thickness of plates = 0.5

Since these plates are placed one above other so total thickness of plates = 0.5 ×50 = 25 cm. 

Total surface area of a cylinder = 2πrh + 2πr²

8. 25 circular plates each of radius 10.5cm nd thickness 1.6cm are placed one above the other to form a solid circular cylinder. Find the curved surface area and volume of cylinder so formed ?

Solution

Given that 25 circular plates each with radius (r) = 10.5 cm 

Thickness = 1.6cm 

Since plates are placed one above other so its height becomes = 1.6 × 25 = 40 cm 

Volume of cylinder = πr²h

9. A path 2m wide surrounds a circular pond of diameter 4cm. How many cubic meters of gravel are required to grave the path to a depth of 20cm .

Solution

Diameter of circular pond = 40m

Radius of pond (r) = 20m.

Thickness = 2m 
Depth = 20cm = 0.2m 
Since it is viewed as a hollow cylinder

Thickness(t) = R - r 
2 = R - r 
2 = R - 20 
R = 22 m

10. A  16m deep well with diameter 3.5m is dug up and the earth from it is spread evenly to form a platform 27.5 m by 7m. find height of platform ?

Solution

Let as assume well is a solid right circular cylinder 

Radius of cylinder (r) = 3.5/2 = 1.75 m 

Height (or) depth of well = 16m.

Volume of right circular cylinder = πr²h

11. A well of diameter 2m is dug 14m deep. The earth taken out of it is spread evenly all around it to form an embankment of height 40cm. Find width of the embankment ? 

Solution

Let us assume well as a solid circular cylinder 

Radius of circular cylinder  = 2/2 = 1 m

Height (or) depth of well = 14m 
Volume of solid circular cylinder = πr²h

12. Find the volume of the largest right circular cone that can be cut out of a cube where edge is 9cm?

Solution

Given that side of cube  = 9cm 

Given that largest cone is curved from cube 

Diameter of base of cone = side of cube 

⇒ 2x = 9 
⇒ r = 9/2 cm 

Height of cone  = side of cube 

⇒ Height of cone(h) = 9cm 

13. A cylindrical bucket, 32 cm high and 18cm of radius of the base, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24cm, find the radius and slant height of the heap. 

Solution

Given, a cylindrical bucket is filled with sand.

Cylindrical bucket has a height 32 cm and a base radius of 18 cm.

Bucket is emptied on the ground and a conical heap of sand is formed.

Height of the conical heap of sand is 24 cm

We have to find the radius and slant height of the heap.

Volume of cylindrical bucket = πr²h

Given, r = 18 cm

h = 32 cm

Volume of sand in cylindrical bucket = (22/7)(18)²(32)

= 32585.14 cm³

Volume of cone = (1/3)πr²h

Given, h = 24 cm

= (1/3)(22/7)r²(24)

= 25.14r² cm³

Given, volume of sand in a cylindrical bucket = volume of conical heap of sand.

25.14r² = 32585.14

r² = 32585.14/25.14

r² = 1296.14

Taking square root,

r = √1296.14

r = 36 cm

Therefore, the radius of the conical heap of sand is 36 cm.

Slant height, l = √r² + h²

= √(36)² + (24)²

= √1296 + 576

= √1872

= 43.27 cm

Therefore, the slant height of the conical heap of sand is 43.27 cm.

14. Rain water, which falls on a flat rectangular surface of length 6cm and breath 4m is transferred into a cylindrical vessel of internal radius 10cm. What will be the height of water in the cylindrical vessel if a rainfall of 1cm has fallen?

Solution

Given length of rectangular surface = 6cm 

Breath of rectangular surface = 4cm 

Height (h) = 1cm 

Volume of a flat rectangular surface = lbh 

6000 × 400 × l 

15. A conical flask is full of water. The flask has base radius r and height h. The water is proved into a cylindrical flask off base radius one. Find the height of water in the cylindrical flask ? 

Solution 

Given base radius of conical flask be r 

Height of conical flask is h 

16. A rectangular tank 15m long and 11m broad is required to receive entire liquid contents from a full cylindrical tank of internal diameter 21m and length 5m. Find least height of tank that will serve purpose? 

Solution

Given length of rectangular tank = 15m 

Breath of rectangular tank = 11m 
Let height of rectangular tank be h 

Volume of rectangular tan k = lbh 

17. A hemisphere tool of internal radius 9cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottles each of diameter 3cm and height 4cm. How many bottles are necessary to empty the bowl. 

Solution

Given that internal radius of hemisphere bowl = 90m 

18. The diameters of the internal and external surfaces of a hollow spherical shell are 6cm and 10cm respectively. If it is melted and recast and recast into a solid cylinder of diameter 14cm, find the height of the cylinder. 

Solution

Internal diameter of hollow spherical shell = 6cm 

Internal radius of hollow spherical shell = 6/2 = 3cm 

External diameter of hollow spherical shell = 10cm 

External radius of hollow spherical shell = 10/2 = 5cm 

Diameter of cylinder = 14cm 

Radius of cylinder = 14/2 = 7cm 

Let height of cylinder = x cm

According to the question 

Volume of cylinder = Volume of spherical shell 

∴Height off cylinder = 8/3 cm 

19.A hollow sphere of internal and external diameter 4cm and 8cm is melted into a cone of base diameter 8 cm. Calculate height of cone ?

Solution

Given internal diameter of hollow sphere (r) = 4cm 

External diameter (R) = 8cm 

20. A cylindrical tube or radius 12cm contains water to a depth of 20cm. A spherical ball is dropped into the tube and the level of the water is raised by 6.75cm. Find the radius of the ball ?

Solution

Given that radius of a cylindrical tube (r) = 12cm 

Level of water raised in tube (h) = 6.75cm 

Volume of cylinder = πr²h

21. 500 persons have to dip in a rectangular tank which is 80m long and 50m broad. What is the rise in the level of water in the tank, if the average displacement of water by a person is 0.04m³ ?

Solution

Given that length of a rectangular tank (r) = 80m 
Breath of a rectangular tank (b) = 50m 
Total displacement of water in rectangular tank 

By 500 persons = 500 × 0.04 m³ 
= 20m³ ...(1)
Let depth of rectangular tank be h 

Volume of rectangular tan k = lbh 

h = 0.5cm 

∴ Rise in level of water in tank (h) = 0.05cm. 

22. A cylindrical jar of radius 6cm contains oil. Iron sphere each of radius 1.5cm are immersed in the oil. How many spheres are necessary to raise level of the oil by two centimetres? 

Solution

Given that radius of a cylindrical jar (r) = 6cm 

Depth/height of cylindrical jar (h) = 2cm 

Let no of balls be 'n' 

Volume of a cylinder =  πr²h

23. A hollow sphere of internal and external radii 2cm and 4cm is melted into a cone of base radius 4cm. Find the height and slant height of the cone? 

Solution

Given that internal radii of hollow sphere(r) = 2cm 

External radii of hollow sphere (R) = 4cm 

24. The internal and external diameters of a hollow hemisphere vessel are 21cm and 25.2cm. The cost of painting 1cm² of the surface is 10 paise. Find total cost to paint the vessel all over? 

Solution

25. A cylindrical tube of radius 12cm contains water to a depth of 20cm. A spherical ball of radius 9cm is dropped into the tube and thus level of water is raised by h cm. What is the value of h? 

Solution

26. The difference between outer and inner curved surface areas of a hollow right circular cylinder 14cm long is 88cm². If the volume of metal used in making cylinder is 176 cm². Find the outer and inner diameters of the cylinder? 

Solution 

Given height of a hollow cylinder = 14 cm 

Let internal and external radii of hollow 

Cylinder be 'r' and R 

Given that difference between inner and outer 

Curved surface = 88cm² 

Curved surface area of cylinder (hollow) 

27. Prove that the surface area of a sphere is equal to the curved surface area of the circumference cylinder?

Solution

Let radius of a sphere be r 

Curved surface area of sphere = 4πr² 

28. The diameter of a metallic sphere is equal to 9cm. It is melted and drawn into a long wire of diameter 2mm having uniform cross-section. Find the length of the wire ? 

Solution

Given diameter of a sphere (d) = 9cm 

Radius (r) = 9/2 = 4.5cm 

29. An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is 1/4 of the radius of the original ball, how many such balls are made ? Compare the surface area, of all the smaller balls combined together with that of the original ball. 

Solution

Given that radius of each of smaller ball = 1/4 Radius of original ball. 

Let radius of smaller ball be r. 

Radius of bigger ball be 4r

∴ Total surface area of small balls is equal to 4 times surface area of big ball.

30. A tent of height 77 dm is in the form a right circular cylinder of diameter 36m and height 44dm surmounted by a right circular cone. Find the cost of canvas at Rs. 3.50 per m² ?

Solution 

Given that height of a tent = 77dm
Height of cone = 44 dm

Height of a tent without cone = 77 - 44 = 33dm = 3.3m

Given diameter of cylinder(d) = 36m 

Radius(r) 36/2 = 18m 

Let 'l' be slant height of cone 

31. Metal spheres each of radius 2cm are packed into a rectangular box of internal dimension 16cm × 8cm × 8cm when 16 spheres are packed the box is filled with preservative liquid. Find volume of this liquid ? 

Solution

Given radius of metal spheres = 2cm 

32. The largest sphere is to be curved out of a right circular of radius 7cm and height 14cm. Find volume of sphere ? 

Solution

Given radius of cylinder (r) = 7cm 

Height of cylinder (h) = 14cm 

Largest sphere is curved out from cylinder 

Thus diameter of sphere = diameter of cylinder 

Diameter of sphere (d) = 2 × 7 = 14cm 

33. A copper sphere of radius 3cm is melted and recast into a right circular cone of height 3cm. Find radius of base of cone ? 

Solution

Given radius of sphere = 3cm 

34. A vessel in the shape of cuboid contains some water. If these identical spheres are immersed in the water, the level of water is increased by 2cm. If the area of base of cuboid is 160cm² and its height 12cm, determine radius of any of spheres ?

Solution

Given that area of cuboid = 160cm² 

35. A copper rod of diameter 1 cm and length 8cm is drawn into a wire of length 18m of  uniform thickness. Find thickness of wire ?

Solution

36. The diameters of internal and external surfaces of hollow spherical shell are 10 cm and 6 cm respectively. If it is melted and recast into a solid cylinder of length of 2 (2/3)cm, find the diameter of the cylinder. 

Solution

Given diameter of internal surfaces of a hollow spherical shell = 10cm 

37. A right angled triangle whose sides are 3 cm, 4 cm and 5 cm is revolved about the sides

containing the right angle in two days. Find the difference in volumes of the two cones so

formed. Also, find their curved surfaces.

Solution

38. How many coins 1.75 cm in diameter and 2mm thick must be melted to form a cuboid 11cm × 10cm × 75cm?

Solution

Given that dimensions of a cuboid 11cm × 10cm × 75cm 

39. A well with inner radius 4 m is dug 14m deep earth taken out of it has been spread evenly

all around a width of 3m it to form an embankment. Find the height of the embankment?

Solution

Given that inner radius of a well (a) = 4m
Depth of a well (h) = 14m 

40. Water in a canal 1.5m wide and 6m deep is flowering with a speed of 10 km/hr . How much area will it irrigate in 30 minutes if 8cm of standing water is desired?

Solution

Given that water is flowering with a speed = 10 km/hr 
In  30 minutes length of flowering standing water = 10× 30/60 km 

= 5 km = 5000 m. 

Volume of flowering water in 30 minutes 

V = 5000 × width × depth m³ 

Given width of canal = 1.5m

Depth of canal = 6m 
V = 5000 × 1.5 ×6 m³ 

V = 45000 m³ 

Irrigating area in 30 minutes if 8cm of standing water is desired = 45000/0.08

= 45000/0.08 = 562500m² 

∴ Irrigated area in 30 minutes = 562500 m² 

41. A farmer runs a pipe of internal diameter 20 cm from the canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of

3 km/h, in how much time will the tank be filled?

Solution 

Volume of pipe = Volume of tank
⇒ πr²h = πR²H
⇒ (10/100)² × h = (5)² × 2
⇒ h/100 = 50
⇒ h = 5000m = 5km

Speed of water = 3km/hr
Time taken to fill tank = 5/3 hr = 5/3 × 60min = 100 min

42. A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly

all around it to a width of 4 m to form an embankment. Find the height of the embankment.

Solution 

Given diameter of well = 3m 
Radius of well = 3/2 m = 4 
Depth of well(b) = 14m 
With of embankment  = 4m 

∴ Radius of outer surface of embankment  = 4 + 3/2 = 11/2  m 
Let height of embankment  = hm 

43. The surface area of a solid metallic sphere is 616 cm² . It is melted and recast into a cone of height 28 cm. Find the diameter of the base of the cone so formed (Use it 22/7) 

Solution

Given height of cone (h) = 28cm

44. The difference between the outer and inner curved surface areas of a hollow right circular cylinder 14cm long is 88cm² . If the volume of metal used in making cylinder is  176 cm³ find outer and inner diameters of the cylinder ?

Solution

Given height of a hollow cylinder = 14cm 

Let internal and external radii of hollow 

Cylinder be 'r' and 'R' 

Given that difference between inner and outer curved surface = 88 cm² .

Curved surface area of hollow cylinder = 2π(R - r)h

⇒ 88 = 2π(R - 0)h
⇒ 88 = 2π(R - r)14 
⇒ R - r = 1 ...(1) 

45. The volume of a hemisphere is  2425 (1/2)cm³ . Find its curved surface area ? 

Solution

Given that volume of a hemisphere = 2424 (1/2)cm³ .

46. A cylindrical bucket 32 cm high and with radius of base 18 cm is filled with sand. This bucket is emptied out on the ground and a conical heap of sand is formed. If the height of the conical heap of sand is formed. If the height of the conical heap is 24 cm. find the radius and slant height of the heap ? 

Solution

Given that height of cylindrical bucket (h) = 32 cm 

Radius(r) = 18cm