A metallic sphere of radius 10.5 cm is melted and then recast into small cones each of radius 3.5 cm and height 3 cm. Find how many cones are obtained? 

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Find out how many small cones you can make when you melt a big metal ball with a radius of 10.5 cm and recast it into smaller cones, each with a radius of 3.5 cm and a height of 3 cm.

A metallic sphere of radius 10.5 cm is melted and then recast into small cones each of radius 3.5 cm and height 3 cm. Find how many cones are obtained?

Approximately 126 small cones can be obtained from melting the metallic sphere.

Explanation

We can find the number of cones obtained by comparing the volume of the sphere to the volume of each cone, and then performing a simple division.

Here’s how:

1. Calculate the volume of the sphere:

   • Volume of a sphere = (4/3)πr³, where r is the radius.
   • Plugging in r = 10.5 cm, we get the volume of the sphere as (4/3)π * (10.5 cm)³ ≈ 4851.17 cm³.

2. Calculate the volume of each cone:

   • Volume of a cone = (1/3)πr²h, where r is the base radius and h is the height.
   • Plugging in r = 3.5 cm and h = 3 cm, we get the volume of each cone as (1/3)π * (3.5 cm)² * 3 cm ≈ 38.5 cm³.

3. Divide the volume of the sphere by the volume of each cone to find the number of cones:

   • Number of cones = Volume of sphere / Volume of each cone
   • ≈ 4851.17 cm³ / 38.5 cm³ ≈ 126.

Therefore, approximately 126 small cones can be obtained from melting the metallic sphere.

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Three Dimensional Shapes in Mathematics

In mathematics, three-dimensional (3D) shapes are geometric objects that exist in three spatial dimensions. These shapes have length, width, and height and are commonly encountered in everyday life. Here are some common three-dimensional shapes:

1. Cube:

   • All sides are equal in length.
   • All angles are right angles (90 degrees).
   • Examples include a standard dice or a Rubik’s Cube.

2. Rectangular Prism:

   • Similar to a cube, but not all sides are necessarily equal.
   • All angles are right angles.
   • Examples include a shoebox or a book.

3. Sphere:

   • A perfectly round object.
   • All points on the surface are equidistant from the center.
   • Examples include a basketball or the Earth.

4. Cylinder:

   • Consists of two parallel bases connected by a curved surface.
   • The bases are typically circles.
   • Examples include a soda can or a paper towel roll.

5. Cone:

   • Has a circular base that tapers to a single point (apex).
   • Examples include an ice cream cone or a traffic cone.

6. Pyramid:

   • A polygonal base connected to an apex by triangular faces.
   • The type of polygon at the base determines the type of pyramid (e.g., triangular pyramid, square pyramid).
   • Examples include the Great Pyramid of Giza or a triangular tent.

7. Prism:

   • Similar to a cylinder, but the bases can be any polygon.
   • The height is the distance between the two bases.
   • Examples include a rectangular prism or a triangular prism.

8. Torus:

   • Shaped like a doughnut or a ring.
   • The surface is generated by revolving a circle in three-dimensional space.
   • Examples include a lifebuoy or a bagel.

These shapes are studied in geometry, and their properties, formulas for volume and surface area, and relationships between them are explored in detail. Understanding three-dimensional shapes is essential in various fields, including architecture, engineering, computer graphics, and physics.

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