Question 1: A cone of height and diameter is mounted on a hemisphere of same diameter. Determine the volume of the solid thus formed.

Answer:

Cone: Height Diameter

Hemisphere: Radius

Total volume = volume of the cone + volume of the hemisphere

Question 2: A buoy is made in the form of a hemisphere surmounted by a right cone whose circular base coincides with the plane surface of a hemisphere. The radius of the base of the cone is and its volume is two-thirds of the hemisphere. Calculate the height of the cone and the surface area of the buoy, correct to two places of decimal.

Answer:

Cone: Height Diameter

Hemisphere: Radius

Therefore

Question 3: From a rectangular solid of metal by by a conical cavity of diameter cm and depth is drilled out. Find:

(i) the surface area of remaining solid,

(ii) the volume of remaining solid,

(iii) the weight of the material drilled out if it weighs

Answer:

Rectangular solid: by by

(i) Surface area of the solid = surface are of the rectangular solid – surface are of the base of the cone +curved surface ares of the cone

(ii) Volume = Volume of the solid – Volume of the cone

(iii) Weight of the material drilled

Question 4: A cubical block of side is surmounted by a hemisphere of the largest size. Find the surface area of the resulting solid.

Answer:

Box: Side

Hemisphere: Radius

Total surface area = surface area of the box – surface area of one side + surface area of the hemisphere

Question 5: A vessel is in the form of an inverted cone. Its height is and the radius of its top, which is open, is It is filled with water up to the rim. When lead shots each of which is a sphere of radius are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Answer:

Cone: Height Radius

Lead shot: Radius number of shots

Question 6: A hemispherical bowl has a negligible thickness and the length of its circumference is Find the capacity of the bowl.

Answer:

Circumference

Question 7: Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius

Answer:

Sphere: Radius

Cone: Radius Height

Question 8: The radii of the bases of two solid right circular cones of same height are and respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms of and

Answer:

Cones: Radius Radius Height

Sphere: Radius

Therefore

Question 9: A solid metallic hemisphere of diameter is melted and recast into a number of identical solid cones, each of diameter and height Find the number of cones so formed.

Answer:

Hemisphere: Radius

Cones: Radius Height Number of cones

Therefore

Question 10: A cone and a hemisphere have the same base and the same height. Find the ratio between their volumes.

Answer:

Cone: Radius Height

Hemisphere: Radius