Students and Teachers Forum

1. The external radius of a hollow spherical ball is 7 cm. If the volume of the ball is 913.52 cm3, find the internal radius of the ball.

Given,         External radius of the spherical ball (R) = 7 cm         Volume of the ball (V) = 913.52 cm3. We know,        Volume of the hollow sphere (V) =43  π(R3 – .....

2. The external and internal radii of a spherical shell of iron are 8 cm and 6 cm respectively. Find the volume of the iron contained in the shell.

Given,        External radius of the spherical shell (R) = 8 cm        Internal radius of the spherical shell (r) = 6 cm. We know,        Volume of the spherical shell (V) = .....

3. If the diameter of the sphere is halved, what will be the difference in volume?

Here,       Let the diameter of a sphere be d.       Radius of the sphere (r) = d2 Then,        volume of the sphere (V1) = 43 πr3                 .....

4. Find the radius and the surface area of the sphere if its volume is 4851 cm3.

Given,         Volume of the sphere (V) = 4851 cm3 We know,         Volume of the sphere (V) = 43πr3 Or,   4851 cm3 = 43 ⨉ 227 ⨉ r3 Or,   4851 cm3 = .....

5. A cylinder vessel of radius 12 cm is partly filled with water. A metallic ball of radius 6 cm is dropped into the vessel. If the ball is completely immersed into the vessel, by how much will the surface of the water raises.

Given,        Radius of the vessel (R) = 12 cm        Radius of the metallic ball (r) = 6 cm. Now, Volume occupied by the metallic ball (v) = 43 πr3               .....

6. A cylinder 42 cm has one hemispherical end. If the radius of the hemisphere is 3.5 cm, find the total volume.

Given,         Height of the cylinder (h) = 42 cm         Radius of the hemisphere (r) = 3.5 cm. Here,        Volume of the object(V) = Volume of the cylinder + Volume of hemisphere   .....

7. The thickness of a hollow spherical ball is 4 cm. If the volume of the metal is 1324.19 cm3, find the external radius of the sphere.

Given,          Thickness of the hollow spherical ball (t) = 4 cm           Let the external and internal radius of the sphere be R and r.. Then,          R – r = 4 .....

8. A cylindrical polythene water pipe is 63 cm long. Its outer diameter is 6 cm. If the volume of the pipe is 990 cm3, find the inner diameter of the pipe.

Given,          Length of the pipe (h) = 63 m          Outer diameter (d1) = 6 cm          outer radius (r1) = 3 cm          Volume of the pipe (V) = .....

9. The volume of a cylinder whose height is double the radius is 169.71 cm3. Find the total surface area of the cylinder.

Given,          Volume of the cylinder (V) = 120 cm          Height (h) = 2 radius (r). We know,         Volume of the cylinder (V)     = πr2h   or, 169.71 .....

10. The product of the external and internal radius of a hollow ball is 35 cm2 and their sum is 12 cm. If the volume of the metal is 913.52 cm3, find the thickness of the ball.

Let the external and internal radius of the ball be R and r respectively. Then,   According to the question,          Rr = 35 cm2  and          R + r= 12 cm.       .....

11. A cylindrical polythene water pipe is 6.3 m long. Its outer and inner diameters are 6 cm and 4 cm respectively. Find the volume of the polythene used in making the pipe.

Given,             Length of the cylindrical pipe (h) = 6.3 m             Outer diameter of the pipe (d1) = 6 cm             Outer radius of the pipe (r1) .....

12. The volume of a spherical ball is 523.80 cm3. Find the cost of painting its surface at the rate of Rs. 12 per sq. metre.

Given,          Volume of the sphere (V) = 523.80 cm3          Let r be the radius of sphere. We know,         Volume of sphere (V) = 43 πr3 Or,   523.80 .....

13. A cylinder 9 cm in length and 4 cm in diameter is closed by a hemisphere at each end. Find its volume.

Given,          Length of the cylinder (h) = 9 cm          Diameter of the cylinder (d) = 4 cm          Radius of the cylinder (r) = 2 cm. Since the hemispheres are at .....

14. Find the total surface area and volume of the given cuboid.

 Length (l) = 15 cm    breadth (b) = 13 cm    height (h) = 4 cm
 

Given;         Length of the cuboid (l) = 15 cm         Breadth of the cuboid (b)= 13 cm             Height of the cuboid (h) = 4 cm We know,  Total surface area of .....

15. The curved surface area of the given hemisphere is 2,772 cm2. Find its volume and total surface area. 

(fig. A hemisphere with radius r)

Given,         Curved surface area of the hemisphere (CSA) = 2722 cm2         Let r be the radius of hemi-sphere. We .....

16. An iron pipe 12 cm long has internal radius of 3 cm. The thickness of the iron is 1 cm. Find the volume of iron in the pipe. Find the cost of the pipe if 1 cu. m of iron pipe costs Rs. 0.50.

Given,        Length of the iron pipe (h) = 12 cm         Internal radius of the iron pipe (r2) = 3 cm        Thickness of the iron pipe (t) = 1 cm.        Then, .....

17. There are two solid hemispherical objects made of earth having the same shape and size. If the total surface area of each hemispherical object is 462 cm2, find the surface area of the sphere which is formed by attaching those two hemispherical objects.

Let r be the radius of sphere. Given,           Total surface area of each hemispherical object (TSA) = 462 cm2    Or, 3πr2    = 462 cm2    Or,  πr2    = .....

18. The surface area of a solid sphere made of earth is 616 cm2. Two hemispheres are formed when it is cut into two equal halves. Find the total surface area of each part.

Given,         Surface area of sphere made of earth (TSA) = 616 cm2         Let r be the radius of sphere. We know,        Surface area of sphere (TSA) = 4πr2 Or, 616 cm2   .....

19. A cylindrical well of 7 m diameter at the base and 20 m deep is dug up and the earth excavated in digging is spread to form a rectangular platform of 22 m ⨉ 17.5 m. Determine the height of the platform.

Given,        Diameter of the cylindrical well (d) = 7 m       radius of the cylindrical well (r) = d2  = 3.5 m       Depth of the cylindrical well (h) = 20 m. We know, Volume of the .....

20. The circumference of base of a cone is 44 cm and the sum of its radius and slant height is 32 cm, find the total surface area of the cone.

Here,       The circumference of base (C) = 44 cm       Sum of radius and slant height (r+l) =32 cm Now, Total surface area of the cone is given by,      TSA = πr( r + l)         .....

21. The height and the radius of a cylindrical wood are equal and the curved surface area is 308 cm2. Find the radius of the wood.

Let,      Height of the cylindrical wood = h      Radius of the cylindrical wood = r Then,       According to the question, r = h       Curved surface area (CSA) = 308 cm2     .....

22. The external and internal radii of a spherical shell of iron are 7 cm and 3.5 cm respectively. Find the volume of the iron contained in the shell.

Given,        External radius of the spherical ball (R) = 7 cm        Internal radius of the spherical ball (r) = 3.5 cm. We know,        Volume of the iron contained in spherical shell .....

23. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

Given,          Diameter of the hemisphere (d) = 14 cm          Radius of the hemisphere (r) = 7 cm          Height of the vessel (h) = 13 cm. Now,       .....

24. A cylinder of length 35 cm has hemispherical ends of radius 10.5 cm. Find the total volume.

Given,         Length of the cylinder (h) = 35 cm         Radius of the spherical ends (r) = 10.5 cm. Now,         Volume of the object (V)  = Volume of the cylinder + Volume of two .....

25. The total surface area of a cone is 814 square cm. If the sum of its slant height and the radius of its base is 37 cm, find the slant height of the cone.

Let r be the radius  and l be the slant height. Then, According to the question, r + l = 37cm And, Total surface area of cone = 814 cm2 Or, πr (r+l) = 814 Or, 22 ⨉ r ⨉ 377 = 814 ∴r = 814×7 / .....

26. The total surface area and diameter of the base of a cone are 300π cm2 and 24 cm respectively, find the slant height of the cone.

Here,       Diameter (d) = 24 cm  So, radius(r)  = 12 cm Also,       total surface area of cone = 300π cm2 Or, πr(r+l) = 300π Or, 12 (12+l) = 300 Or, 12 + l = 25 Or, l = 25-12 Or, l = 13 .....

27. In a cone, 24 times the radius is equal to 7 times the height. If the volume of the cone is 3163 cm3, find the measure of the slant height.

Let r be the radius and h be the height of the cone. According to the question,       24r = 7h    ∴r = 724 h And, the volume of the cone = 3163 cm3 Or, 13πr2h = 3163 Or, 13π(7h24)2h = 3163 Or, .....

28. If the volume and vertical height of a cone are 23100 cubic cm and 50 cm respectively, find the diameter of its base.

Here, Let d be the diameter of cone. Volume of the cone = 23100 cm3 Or, 112πd2h = 23100 Or,  112πd2×50 = 23100 Or,   d2 = (23100 X 12/50π) Or, d2 =  1764 Or, d = 42 cm Hence, the diameter of the base of .....

29. Calculate the plane surface and total surface area of a cylinder having the curved surface area of 1848 cm2 in which the sum of radius and height is 35 cm.

Let, r and h be the radius of the base of cylinder and height of the cylinder respectively. According to question,          Sum of radius and height = 35 cm    Or, r + h = 35     ∴   r = .....

30. The upper part of a cylindrical tank is hemispherical. If the height of the tank is 4.5 m and the circumference of its base is 22 m, how many litres of water can the tank hold?

Here,        circumference of base of the cylinder = 22 m Or, 2πr = 22m                      [ r = radius of base.] Or, 2×227×r=22 ∴ r = 3.5 m Height .....

31. Two hemispheres of radius 14 cm are placed at both the ends of a cylinder whose height is 3 times the radius. Find the volume of the solid object.

Here, The radius of the hemisphere = radius of base of cylinder (r) = 14 cm The height of the cylinder (h) = 3r = 3×14 = 42 cm Now,     volume of the solid object = volume of two hemisphere + volume of the cylinder   .....

32. Find the volume of the given solid cone.

Here,     slant height (l) = 25 cm     height (h) = 24 cm Now, radius (r) =l2-h2                =252-242                 = 7 .....

33. A solid is made by a hemisphere and cylinder having equal radii. The volume of the solid is 111804 cm3. If the height of the cylinder is 80 cm, find the total surface area of the solid.

Here,         height of the cylinder (h) = 80 cm         radius of hemisphere = the radius of the cylinder = r (say)        volume of the solid = 111804 cm3 Or, volume of the hemisphere + .....

34. Find the radius and volume of the sphere if its surface area is 16π cm2.

Given,          Surface area of the sphere (TSA) = 16π cm2 We know,          Surface area of the sphere (TSA) = 4πr2 Or,    16π cm2 = 4π ⨉ r2 Or,    r2 = .....

35. Find the surface area and volume of the given hemisphere of radius 14 cm.

Given,          Radius of the hemisphere (r) = 14 cm. We know, Surface area of the hemisphere (TSA) = 3πr2               = 3 ⨉ 3.14 ⨉142 cm2           .....

36. In the figure alongside, the base area of the cylinder is 100 cm2 and its height is 3 cm. If the volume of the whole solid is 500 cm3, find the height of the solid thus formed.

Here, Height of cylinder = 3 cm Area of base = 100 cm2 We know, Volume of cylinder = Base Area ⨉ height                                 = 100 ⨉ 3 cm3     .....

37. The height and two bases of trapezium are 8cm, 6 cm and 7 cm respectively. Find its area.

Given,    Height (h) = 8 cm    Bases, b1 = 6 cm  & b2 = 7 cm    Area = ? We have, Area of the trapezium    = 12 h (b1 + b2)                 .....

38. In a right-angled isosceles triangular land, one of the equal edges is 10 m. Find the area of the land and the remaining edges.

Here, Let, the base and the perpendicular of the right-angled triangular land be "a". Then, a = 10 cm Now, Area of the right-angled triangle is given by, Area = 12 ⨉ p ⨉ b          = 12 ⨉ a ⨉ .....

39. The perimeter and the length of two sides of a triangle are 36 cm, 8 cm, and 12 cm respectively. Find the area of the triangle.

Let, a, b and c be the three sides of the triangle. Given ,       a = 8 cm        b = 12 cm And,       P =  a + b + c = 36 cm Or, c = P - ( a+b)         = 36 - (8 + .....

40. A triangular land having area 336 m2 and perimeter 84 m has length of an edge 26 m. Calculate the measure of remaining two sides.

Solution: Let a, b and c are the lengths of 3 sides of triangular land.. Here,         a + b + c = 84 m And, c = 26 m Hence, a + b + 26 cm = 84 m Or, a + b = 58 m……….1). We have, s = (a + b + c)/2 = .....

41. In ∆ABC, if AB = 10 cm, BC = 24 cm and AC = 30 cm, find the area of ∆ABC.

Given, AB (c) = 10 cm    BC (a) = 24 cm and  AC (b) = 30 cm Now, Semi-perimeter of ∆ABC (s) = 12 (a + b + c)                                 .....

42. A triangle having area √135 sq. meter and perimeter 18 m has a side 4 m. Find the measure of remaining two sides.

Let a, b and c are the lengths of 3 sides of triangle. Here,          a + b + c = 18 m And, c = 4 m Hence, a + b + 4m = 18 m Or,   a + b = 14 m……….(1) We have, Semi-perimeter (s) = 12(a + b + .....

43. The area of an equilateral triangular land is 9003 square meter. Find the length of each side.

Here,       Let, a be the side of equilateral triangle.       Area of an equilateral triangular land = 9003 m2 Using formula,      34a2 = 9003 m2 Or, a2 = 900 ⨉ 4 m2 Or, a = 900⨉4 .....

44. What is the area of the given triangle ABC?

Here,        a = 5cm,        b = 13cm and        c = 12cm Now, semi-perimeter (s) = 12 (a + b + c)                         .....

45. The area of an equilateral triangle is 163 cm2, find its perimeter.

Here,        Let a be the length of sides of the equilateral triangle, Given,       Area of an equilateral triangle = 163 cm2 Or, 34a2 =163cm2 Or, a2=16 ⨉ 4 Or, a =16 ⨉ 4       .....

46. What is the area of the given triangle ABC?

Let,   a = 5 cm,   b = 13 cm   and c = 12 cm Now, s = 12 (a + b + c)    = 12 (5 + 13 + 12) cm    = 15 cm The area of the ∆ABC is given by,      Area = .....

47. Find the area of given plane figure.

Here, In right angled triangle ADC, DC = √(AC2 - AD2)        = √(5.32 - 52)        = 1.76 cm Area of triangle ADC = 12 ⨉ AD ⨉ CD           .....

48. The area of the equilateral triangle is 9√3 sq. cm. Find its side.

Let a be the side of the given equilateral triangle. We have,         Area of equilateral triangle = √34 a2     Or, 9√3 = √34a2     Or, a2 = 36 cm2     .....

49. The base and height of a parallelogram are 6 cm and 8 cm. Find the area.

We have given,       Base (b) = 6cm       Height (h) = 8cm       Area = ? We know,     Area of the parallelogram= b×h                     .....

50. Find the area of right-angled triangle ABC; given that, AB = 5 cm, BC = 4 cm and CA = 3 cm.

Given,        Base (b) = CA = 3cm        Perpendicular (p) = BC = 4cm.       [ Since , longest side of a right-angled triangle is hypoteneous = 5cm.] We have, Area of right angle triangle .....