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 hemisphere surmounted by a right cone whose circular base coincides with the plane surface of hemisphere. The radius of the base of the cone is and its volume is two-third 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

Total surface area of the solid

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

Therefore:

Question 6: A hemi-spherical bowl has negligible thickness and the length of its circumference is . Find the capacity of the bowl.

Answer:

Circumference

Therefore Volume of the bowl

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

The maximum volume of the cone

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

Ratio of their volumes