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