Notes: Important formuals
Parameters of a Sphere: Radius of the Sphere ( )
Question 1: The surface area of a sphere is find its volume.
Answer:
Question 2: The volume of a sphere is ; find its diameter and the surface area.
Answer:
Question 3: A spherical ball of lead has been melted and made into identical smaller balls with radius equal to half the radius of the original one. How many such balls can be made?
Answer:
Let the radius of the original ball be and the volume be
Similarly, let the radius of the smaller ball be and the volume be
Question 4: How many balls each of radius can be made by melting a bigger ball whose diameter is
Answer:
Diameter of the bigger ball
Therefore the Radius of the bigger ball
Radius of the smaller ball
Question 5: Eight metallic spheres; each of radius are melted and cast into a single sphere. Calculate the radius of the new sphere.
Answer:
Number of metallic spheres
Radius of the metalic sphere
Let the radius of the new sphere
Question 6: The volume of one sphere is times that of another sphere. Calculate the ratio of their: (i) radii (ii) surface areas
Answer:
Let the volume of the 1st sphere be and its radius be
Similarly, Let the volume of the 2nd sphere be
and its radius be
(ii) Let the surface area of the two spheres be and
respectively.
Question 7: If the number of square centimeters on the surface of a sphere is equal to the number of cubic centimeters in its volume, what is the diameter of the sphere?
Answer:
Let the radius of the sphere
Given, Surface area = Volume of the sphere (only in terms of numerical value)
Therefore Diameter
Question 8: A solid metal sphere is cut through its center into equal parts. If the diameter of the sphere is
find the total surface area of each part correct to two decimal places.
Answer:
Diameter of the sphere
Question 9: The internal and external diameters of a hollow hemispherical vessel are and
respectively. Find: (i) internal curved surface area, (ii) external curved surface area, (iii) total surface area, (iv) volume of material of the vessel
Answer:
Question 10: A solid sphere and a solid hemisphere have the same total surface area. Find the ratio between their volumes.
Answer:
Let the radius of the sphere be and hemisphere be
.. … … (i)
It is given that:
Substituting this in (i) we get
Question 11: Metallic spheres of radii and
respectively are melted and recasted into a single solid sphere. Taking
find the surface area of solid sphere formed.
Answer:
Let the radius of the new sphere
Volume of the new sphere = total volume of the three spheres. Therefore
Question 12: The surface area of a solid sphere is increased by without changing its shape. Find the percentage increase in its: (i) radius (ii) volume.
Answer:
Let the initial Surface area be Radius be
and Volume be
Let the final Surface area be Radius be
and Volume be
This means that there has been an increase of in the radius.
This means that the volume has increased by