Notes: Important formuals

*Parameters of a Sphere:* Radius of the Sphere ()

Volume of the sphere

Surface area of a sphere

Question 1: The surface area of a sphere is , find its volume.

Answer:

Surface area of Sphere

Therefore

Therefore

Volume

Question 2: The volume of a sphere is ; find its diameter and the surface area.

Answer:

Volume of Sphere

Therefore

Surface area

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

Given

Therefore the number of smaller balls

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

Therefore No of Smaller balls

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

Therefore

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 .

Given

(i) Therefore

(ii) Let the surface area of the two spheres be and respectively.

Therefore

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

Radius

Total surface area

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:

Internal Radius

External Radius

(i) Internal surface area

(ii) External surface area

(iii) Total surface area

(iv) Volume of Material

Question 10: A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.

Answer:

Let the radius of the sphere be and hemisphere be

Surface area of sphere

Surface are of hemisphere

Volume of sphere

Volume of hemisphere

Therefore the ratios of their volume … … … (i)

It is given that:

Substituting this in (i) we get

Ratios of their volume

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

Therefore

Surface area

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 .

Given

Therefore

This means that there has been an increase of in the radius.

Taking ratios of the volumes we get

This means that the volume has increased by