"H.O.T.S" based on CONVERSION OF SOLIDS

PROBLEMS BASED ON CONVERSION OF SOLIDS

1.         A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm.  The solid is placed in a cylindrical tub, full of water, in such a way that the whole solid is submerged in water. If the radius of the cylindrical tub is 5 cm and its height is 10.5 cm, find the volume of water left in the cylindrical tub (use  p =22/7)                                        (Ans: 683.83 cm³)

2.         A bucket of height 8 cm and made up of copper sheet is in the form of frustum of right    circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively.  Calculate

i)        the height of the cone of which the bucket is a part

ii)      the volume of water which can be filled in the  bucket

iii)    the area of copper sheet required to make the bucket (Leave the answer in terms of                                                                                    (Ans:  129πcm²)

3.         A sphere and a cube have equal surface areas. Show that the ratio of the volume of the

 sphere to that of the cube is  √6: √π

         

4.         A right triangle whose sides are 15 cm and 20 cm is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed.

(Ans : 3768cu.cm,1318.8   Sq.cm)

5.         Water in a canal 30 dm wide and 12 dm deep is flowing with a velocity of 10 km/h. How much area will it irrigate in 30 minutes if 8 cm of standing water is required for irrigation?                                                                             (Ans: 225000 cu. m)

                                         

6.         A cylindrical vessel of diameter 14 cm and height 42 cm is fixed symmetrically inside a similar vessel of diameter 16 cm and height 42 cm. The total space between two vessels is filled with cork dust for heat insulation purposes. How many cubic centimetres of cork dust will be required?                                                                     (Ans:1980 cu.cm)

7.         An ice-cream cone has a hemispherical top. If the height of the cone is 9 cm and base radius is 2.5 cm, find the volume of ice cream cone.                          (Ans: 91π cu.cm)

8.         A building is in the form of a cylinder surrounded by a hemispherical vaulted dome and contains   41 x19/21  cu m of air. If the internal diameter of the building is equal to its total height above the floor, find the height of the building.                             ( Ans : 4m)

9.         The height of the Cone is 30 cm A small cone is cut of f  at the top by a plane parallel to its base if its volume be 1/27of  the volume of the given cone at what height above the base is the section cut                                                                                  (Ans:20 cm)

10.       A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8/9 th of the curved surface of the whole cone, find the ratio of the line segments into which the cone’s altitude is divided by the plane.

(Ans:1:2)

11.       Two right circular cones X and Y are made X having 3 times the radius of Y  and Y having half the Volume of X.  Calculate the ratio of heights of X and Y.      (Ans: 9 : 2)

12.       If the areas of three adjacent faces of cuboid are x, y, z respectively, Find the volume of the cuboids.                                                                                               

13.       A shuttlecock used for playing badminton has the shape of a frustum of a Cone  mounted on a hemisphere.  The external diameters of the frustum are 5 cm and 2 cm, and the height of the entire shuttlecock is 7cm.  Find the external surface  area.  

                                                                                                                    (Ans: 74.26cm²)   

14.       A Solid toy in the form of a hemisphere surmounted by the right circular cone of height 2cm and diameter of the base 4 cm .If  a right circular cylinder circumscribes  the toy, find how much more space than the toy it will cover.                                (Ans: 8π )

15.       A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed as shown in the figure. What fraction of water flows out.

                                                                                                                                [Ans:3/8 ]

16.       A golf ball has a diameter equal to 4.1cm. Its surface has 150 dimples each of radius 2mm. Calculate the total surface area which is exposed to the surroundings assuming that the dimples are hemispherical.                                                                     (Ans: 71.68)

 

17.       A solid metallic circular cone 20cm height with vertical angle 60 is cut into two parts at the middle point of its height by a plane parallel to the base. If the frustum, so obtained be drawn into a wire of diameter  1/16cm Find the   length of the wire.    (Ans:7964.4m)

                

18.       If the areas of the circular bases of a frustum of a cone are 4cm² and 9cm² respectively and the height of the frustum is 12cm. What is the volume of the frustum. (Ans:44cm²).

         

20.       A vessel in shape of a inverted cone is surmounted by a cylinder has a common radius of 7cm this was filled with liquid till it covered one third the height of the cylinder. If the height of each part is 9cm and the vessel is turned upside down. Find the volume of the liquid and to what height will it reach in the cylindrical part. (Ans:924πcm, 6cm)