A block of metal measuring \(3.0 \mathrm{~cm} \times 4.0 \mathrm{~cm} \times 5.0 \mathrm{~cm}\) has a mass of \(470.0\) g. What is the density of the metal in grams per cubic centimeter?

The volume of a rectangular prism is a measure of how much space it occupies. Imagine filling a box with unit cubes; the volume tells you how many cubes you can fit inside. For any rectangular prism, you can find the volume by multiplying the length, width, and height together. This can be represented by the formula:
\[V = l \times w \times h\]
In the context of the provided exercise, you have a metal block with dimensions of 3.0 cm by 4.0 cm by 5.0 cm. So, its volume is calculated by multiplying these dimensions:\[V = 3.0 \, \text{cm} \times 4.0 \, \text{cm} \times 5.0 \, \text{cm} = 60.0 \, \text{cm}^3\]
This result represents the total volume of the metal block, indicating it would occupy 60 cubic centimeters if placed in a container.

The mass-to-volume ratio is an essential concept in understanding various physical properties, particularly the density of an object. As the name implies, this ratio compares the amount of mass an object has to the space it occupies. To find it, you simply divide the total mass by the total volume.
In practice, a high mass-to-volume ratio suggests that an object is dense, meaning a lot of material is packed into a small space. Conversely, a low ratio indicates that an object is less dense or more spread out. In the exercise, the metal block has a mass of 470 g and occupies a volume of 60 cm³, hence its mass-to-volume ratio is used to calculate its density.

Density is a property that quantifies how compact the mass in a substance or object is. The density formula is remarkably straightforward: \[\rho = \frac{mass}{volume}\]
Where \(\rho\) (the Greek letter rho) represents density, 'mass' is the total mass of the object, and 'volume' is how much space the object takes up. To find density, divide the mass (usually in grams) by the volume (typically in cubic centimeters). This formula is instrumental in comparing different materials or substances to determine which one is heavier for the same volume. In our metal block example, the density is calculated from its mass of 470 g and a volume of 60 cm³, yielding the metal's density.

Density is measured in units that reflect the mass-to-volume relationship. In SI (International System of Units), density is typically given in kilograms per cubic meter (kg/m³), but for smaller objects or laboratory measurements, grams per cubic centimeter (g/cm³) are often used.
It's crucial to match the mass and volume units when calculating density. If you have mass in grams and volume in cubic centimeters, the density will be in g/cm³, which tells you how many grams of substance are in one cubic centimeter. The metal's density from our exercise is 7.83 g/cm³, a unit that is particularly intuitive for imagining the weight of small objects.