How Many Spherical Balls of Radius 1 cm Can Be Made from a Cube with a Side Length of 22 cm?

Introduction

In the field of geometry and mathematics, the problem of determining how many spherical balls can be made from a given volume often arises. This question is not only academically interesting but also practical in various applications, such as manufacturing processes and material science. This article will walk you through the process of calculating how many spherical balls with a radius of 1 cm can be made from a cube with a side length of 22 cm.

Understanding the Problem

The main task is to calculate the number of spherical balls with a given radius (1 cm) that can fit inside a cube of a specified side length (22 cm). To do this, we need to calculate the volumes of both the cube and the sphere and then divide the volume of the cube by the volume of one sphere.

Calculating the Volume of the Cube

First, we calculate the volume of the cube using the formula:

Volume of a cube side length3

Substituting the given side length (22 cm):

Volume of the cube 223 10648 cm3

Calculating the Volume of One Spherical Ball

The volume of a sphere is given by the formula:

Volume of a sphere (frac{4}{3} pi r^3)

Substituting the radius (1 cm) and using (pi frac{22}{7}):

Volume of one spherical ball (frac{4}{3} times frac{22}{7} times 1^3 frac{4}{3} times frac{22}{7} 4.19047619 text{ cm}^3)

Calculating the Number of Spherical Balls

To find out how many spherical balls can be made, we divide the volume of the cube by the volume of one spherical ball:

Number of spherical balls (frac{Volume of the cube}{Volume of one spherical ball} frac{10648}{4.19047619} approx 2545.66)

Since the number of balls must be a whole number, we round down to the nearest integer, resulting in 2545 spherical balls.

Note: This calculation assumes perfect packing and no waste. In real-world scenarios, the actual number of balls that can be packed into the cube might be less due to the inefficiencies of packing spheres.

Conclusion

In summary, a cube with a side length of 22 cm can be used to make approximately 2545 spherical balls with a radius of 1 cm, assuming perfect packing. This calculation helps in understanding the relationship between the volume of a cube and the volume of spherical balls, which is essential in various fields such as engineering, manufacturing, and mathematics.

Key Takeaways:

Volume of the Cube: 223 10648 cm3 Volume of One Spherical Ball: (frac{4}{3} times frac{22}{7} times 1^3 approx 4.19047619 text{ cm}^3) Number of Spherical Balls: (frac{10648}{4.19047619} approx 2545) (rounded down to the nearest integer)