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The calculation of volume is a fundamental concept in geometry and physics, especially when dealing with objects of regular shape. Volume measures how much space an object occupies. To calculate volume, different formulas apply to different shapes. For example:
- **Cube**: The volume of a cube is calculated by taking the cube of its side length. Mathematically, this is given by \( V_{ ext{cube}} = s^3 \), where \( s \) is the side length. For our cube with a side length of 5.0 cm, the volume is \( 125.0 \text{ cm}^3 \).
- **Cylinder**: The volume of a cylinder is given by \( V_{ ext{cylinder}} = \pi r^2 h \), where \( r \) is the radius of the base and \( h \) is the height. A drilled cylindrical hole with a radius of 1.0 cm and height of 5.0 cm results in a volume of \( 5\pi \text{ cm}^3 \). Understanding these relationships helps in finding how much material makes up an object or how much it can contain.

Mass and weight are often confused but are distinct concepts in physics. **Mass** is a measure of how much matter an object contains and is usually measured in grams or kilograms. It does not change based on location.
**Weight**, on the other hand, is the force exerted by gravity on an object. It is calculated using the formula \( F = mg \), where \( F \) is the force (or weight) in newtons, \( m \) is the mass in kilograms, and \( g \) is the acceleration due to gravity, typically \( 9.8 \text{ m/s}^2 \) on Earth. After drilling the hole, the cube's mass was determined using its weight and rearranging the formula: \( m = \frac{F}{g} \). Converting between mass and weight is crucial for understanding how they interact under gravitational forces.

A cylinder is a three-dimensional shape with two parallel circular bases and a height. Calculating the volume of a cylinder is key in many scientific and engineering problems. The formula \( V_{ ext{cylinder}} = \pi r^2 h \) captures this, involving:

• **\( \pi \)**: A constant approximately equal to 3.14159.
• **\( r \)**: The radius of the cylinder's base.
• **\( h \)**: The height of the cylinder.

In our exercise, the cylindrical hole has a radius of 1.0 cm and a height of 5.0 cm, resulting in a volume of \( 5\pi \text{ cm}^3 \). This measurement helps in determining how much of the original cube’s material is removed and provides insight into the configuration of objects.

Density is a measure of how compact the mass in a substance is. It is commonly expressed in units such as grams per cubic centimeter (g/cm³). For metals, knowing the density is important for their identification and use in construction and manufacturing.
The formula for calculating density is \( \rho = \frac{\text{mass}}{\text{volume}} \). In the exercise, the metal density of the modified cube was calculated by substituting the cube's mass (765.3 g) and volume after the hole was drilled (109.28 cm³). This led to a density of approximately 7.00 g/cm³.
Understanding metal density aids in tasks like calculating weight for shipping, ensuring material strength, and selecting the right metal for applications based on its density.