Please refer to the following lecture notes for the formulas used in this exercise: Notes
Question 1: A solid sphere of radius is melted and recast into solid right circular cones of radius
and height
Calculate the number of cones recast. [2013]
Answer:
Sphere: Radius
Cone: Radius and Height
Question 2: A hollow sphere of internal and external diameters and
respectively is melted into a cone of base diameter
Find the height of the cone. [2002]
Answer:
Internal diameter Internal radius
External diameter External radius
Radius of the
Question 3: The radii of the internal and external surfaces of a metallic spherical shell are and
respectively. It is melted and recast into a solid right circular cone of height
Find the diameter of the base of the cone.
Answer:
Internal radius
External radius
Height of the cone
Therefore and hence diameter
Question 4: Total volume of three identical cones is the same as that of a bigger cone whose height is and diameter
Find the radius of the base of each smaller cone, if height of each is
Answer:
Let the radius of the cone
Height of the cone
Bigger cone: Height and radius
Question 5: A solid rectangular block of metal by
by
is melted and formed into a solid sphere. Calculate the radius of the sphere.
Answer:
Dimension of the block
Let the radius of the sphere
Question 6: A hemispherical bowl of internal radius is full of liquid. This liquid is to be filled into conical shaped small containers each of diameter
and height
How many containers are necessary to empty the bowl?
Answer:
Bowl: Internal radius
Cone: radius , Height
Question 7: A hemispherical bowl of diameter is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius
Find the height of the cone if it is completely filled. [2010]
Answer:
Hemisphere: Radius
Cone: Radius , Height
Question 8: A solid cone of radius and height
is melted and made into small spheres of radius
Find the number of spheres formed. [2011]
Answer:
Cone: Radius , Height
Sphere: Radius
Question 9: The total area of a solid metallic sphere is It is melted and recast into solid right circular cones of radius
and height
Calculate: (i) the radius of the solid sphere, (ii) the number of cones recast. [2000]
Answer:
Surface area
(i)
(ii) Cone: Radius , Height
Question 10: A solid metallic cone, with radius and height
, is made of some heavy metal A. In order to reduce its weight, a conical hole is made in the cone as, shown and it is completely filled with a lighter metal B. The conical hole has a diameter of
and depth
calculate the ratio of the volume of metal A to the volume of the metal B in the solid.
Answer:
Cone: Radius , Height
Conical hole: Radius , Height
Question 11: A hollow sphere of internal and external and
respectively is melted and recast into small cones of base radius
and height
Find the number of cones. [2012]
Answer:
Sphere: Internal radius , External radius
Cone: Radius , Height
Question 12: The surface area of a solid metallic sphere is It is melted and recast into solid right circular cones of radius
and height
Calculate: (i) the radius of the sphere (ii) the number of cones recast. (Take
) [2014]
Answer:
Surface area of sphere
Cone: Radius , Height
Hence