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.

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.

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)