A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, of diameter 3.5 cm and height 3 cm . The number of cones so formed is

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From sphere to cones: Experience the mathematical alchemy as a 21 cm diameter metallic sphere undergoes a transformation, yielding numerous smaller cones, each measuring 3.5 cm in diameter and 3 cm in height.

A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, of diameter 3.5 cm and height 3 cm . The number of cones so formed is

The number of cones formed is 504.

Given:

• Diameter of the sphere = 21 cm
• Diameter of the cone = 3.5 cm
• Height of the cone = 3 cm

First, let’s find the volume of the sphere:

Volume of sphere = (4/3) * π * r^3

= (4/3) * π * (10.5)^3

= (4/3) * π * 1157.625

= 4851 cm^3

Now, let’s find the volume of one cone:

Volume of cone = (1/3) * π * r^2 * h

= (1/3) * π * (1.75)^2 * 3

= (1/3) * π * 9.765625 * 3 = 9.625 cm^3

Now, to find the number of cones formed, we divide the volume of the sphere by the volume of one cone:

Number of cones = Volume of sphere / Volume of one cone = 4851 / 9.625 ≈ 504

So, the number of cones formed is 504.

Shape Conversion of Solids

Shape conversion of solids refers to the process of transforming one solid geometric shape into another while preserving certain characteristics such as volume or surface area. This concept is commonly encountered in geometry and engineering applications where it’s essential to understand how shapes can be transformed or manipulated.

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There are various methods and techniques for shape conversion depending on the specific properties and requirements of the shapes involved. Some common techniques include:

1. Scaling: This involves uniformly changing the size of a solid shape while preserving its overall proportions. For example, scaling a cube uniformly will result in another cube with different dimensions.

2. Stretching and Compression: These operations involve changing the dimensions of a solid shape along one or more axes. For example, stretching a cube along one axis will result in a rectangular prism.

3. Shearing: Shearing involves shifting one part of a solid shape relative to another part along a specific axis. This can result in a change in the shape of the solid.

4. Rotation: Rotating a solid shape involves turning it around a specific axis. This can result in different orientations of the same shape.

5. Deformation: Deformation involves altering the shape of a solid by bending, twisting, or otherwise changing its structure. This is common in materials science and engineering applications.

6. Combination of Transformations: Often, complex shape conversions involve a combination of different transformations. For example, a shape may be scaled, rotated, and then deformed to achieve the desired result.

In engineering and design, shape conversion is crucial for tasks such as morphing structures, optimizing designs for specific purposes (e.g., aerodynamics), and simulating physical processes. Additionally, understanding shape conversion helps in various computational tasks such as finite element analysis, computer-aided design (CAD), and computer graphics.

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