Find the density of cuboid of dimensions \(3 \mathrm{~cm} \times 5 \mathrm{~cm} \times 7 \mathrm{~cm}\) and having mass \(1 \mathrm{~kg}\) in SI system.

Understanding how to find the volume of a cuboid is essential in many areas of mathematics and science. The volume of a cuboid is found by multiplying its length, width, and height together. This follows the general formula:

\[Volume = length \times width \times height\]

The exercise given requires calculating the volume of a cuboid with dimensions 3 cm by 5 cm by 7 cm. Multiplying these dimensions together yields:
\[Volume = 3 \mathrm{~cm} \times 5 \mathrm{~cm} \times 7 \mathrm{~cm} = 105 \mathrm{~cm^3}\]

Visualizing this can make it easier to understand. If you imagine a cube with a side of 1 cm, it has a volume of 1 cm³. A cuboid is like several of these cubes put together. Here, we have a shape that would fit 105 of these small cubes, hence its volume is 105 cm³. Remembering that volume measures the space an object occupies can make it easier to grasp this three-dimensional concept.

To understand the mass to density conversion, one must first recognize that density is defined as mass per unit volume. This is given by the formula:

\[Density = \frac{Mass}{Volume}\]

In our exercise, the cuboid's mass is given as 1 kg, and we’ve already calculated its volume to be 105 cm³. Using the formula, we perform the conversion by dividing the mass by the volume:

\[Density = \frac{1 \mathrm{~kg}}{105 \mathrm{~cm^3}} = \frac{1}{105} \mathrm{kg/cm^3}\]

Recognizing that density is the amount of matter in a given space, it becomes clear that as the volume of an object increases with a constant mass, the density decreases, and vice versa. Thus, this cuboid with a mass of 1 kg occupies 105 cm³ of space, and its density reflects how much mass there is per unit of volume in its solid form.

Working with SI (International System of Units) is crucial as it is the most widely used system of measurement around the world. When dealing with density, the standard SI unit is kilograms per cubic meter (kg/m³). In order to convert our density from kilograms per cubic centimeter (kg/cm³) to kilograms per cubic meter (kg/m³), we need to understand the conversion factors:

For length conversions:
\[1 \mathrm{~cm} = 0.01 \mathrm{~m}\]

This implies a cube that is 1 cm³ in volume is equivalent to 0.000001 m³ because:

\[1 \mathrm{cm^3} = (0.01 \mathrm{~m})^3 = 0.000001 \mathrm{m^3}\]

To convert our calculated density into SI units, we multiply it by \(10^6\) -- this is because \(10^6\) cm³ equals 1 m³. So, the density in SI units from our exercise is:

\[Density_{SI} = Density \times Conversion~Factor = \frac{1}{105} \mathrm{kg/cm^3} \times 10^6 \mathrm{kg/m^3}\]

After performing this calculation, we find the density to be approximately 9523.81 kg/m³. Knowing how to convert to and use SI units is vital as these measurements provide a common language for scientists all over the world, allowing for clear communication and comparison of data.