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
Therefore number of cones re-casted
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
Volume of metal A
Volume of metal B
Therefore Ratio
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
(i)
Hence
(ii)