91Ó°ÊÓ

When calculating the volume of a sphere, the mathematical formula used is \[ V = \frac{4}{3} \pi r^3 \] where \(V\) is the volume, \(\pi\) is a constant (approximately 3.14159), and \(r\) is the radius of the sphere.
The radius is half the diameter. Therefore, if given a diameter, simply divide it by two to obtain the radius. For instance, if the diameter of a table tennis ball is 3.75 cm, the radius would be:
\[ r = \frac{3.75}{2} = 1.875 ext{ cm} \]
Once the radius is known, substitute it back into the formula to obtain the volume:\[ V = \frac{4}{3} \pi (1.875\text{ cm})^3 \]
This provides the volume of a single ball.Always ensure that the units are consistent, especially when applying this formula in practical contexts.

The Earth can be considered a giant sphere. To find its surface area, the following formula for the surface area of a sphere is used:\[ A = 4 \pi R^2 \]where \(A\) represents the surface area, \(\pi\) is the mathematical constant, and \(R\) is the radius of the Earth.
The average radius of Earth is approximately 6,371 kilometers. For scientific calculations, this is often converted to centimeters:\[ R = 6,371,000 ext{ m} \times 100 \text{ cm/m} = 6.371 \times 10^8 ext{ cm} \]
Plug this radius back into the formula for surface area:\[ A = 4 \pi (6.371 \times 10^8 ext{ cm})^2 \]
This calculation results in a vast number, depicting the enormous scale of Earth's surface. Understanding this helps when relating geometrical problems to real-world entities like Earth.

Avogadro's number is a fundamental constant in chemistry and physics, defined as \(6.022 \times 10^{23}\). It represents the number of particles in one mole of substance, often atoms or molecules.
In practical terms, Avogadro's number indicates that this staggering amount of identical objects can fill a space under given conditions.
In our table tennis ball problem, Avogadro's number is used to model the total count of balls:

• Each ball has a defined volume, calculated using the sphere formula.
• We assume to have \(6.022 \times 10^{23}\) such balls, each separated by a small space making the real effective volume larger by 25%.

This can be extrapolated to numerous applications beyond just counting particles, providing insights into both microscopic and macroscopic scales.

Geometrical volume expansion typically refers to how objects increase in effective volume due to spacing between them or other factors.
In the context of our exercise, it refers to the 25% increase in volume calculated for the arrangement of table tennis balls.

• Each ball has a specific volume measured by the sphere's formula.
• The spaces between them effectively increase their collective volume by 25%.

Mathematically, this is expressed by multiplying the total volume obtained by the individual balls by 1.25:\[ V_{total\_space} = 1.25 \times (\text{total volume of balls}) \]This concept emphasizes how adding space, even in tight packings of spherical objects, leads to more considerable expansion than what might initially be perceived.