The densiry of diamond is \(3.51 \mathrm{~g} \cdot \mathrm{cm}^{-3}\). The international (hut non-SI) unit for weighing diamonds is the "carat," with 1 carat \(=200 \mathrm{mg}\). What is the volume of a \(0.300\)-carat diamond?

Understanding the conversion from carats to grams is essential when dealing with precious stones like diamonds.
The carat is a unit of mass that is typically used to measure gemstones and pearls, with one carat being equivalent to 200 milligrams. While this unit is convenient for the gem industry, scientists and many other industries use grams as a standard unit of mass.
To convert carats to grams, simply multiply the number of carats by 0.2, since there are 1000 milligrams in a gram and a carat is 200 milligrams. For a gemstone that is 0.300 carats, the conversion would be:
\[\[\begin{align*}\text{Weight in grams} &= 0.300 \text{ carats} \times 0.2 \frac{\text{g}}{\text{carat}} \&= 0.060 \text{ grams}.\end{align*}\]\]This step is the foundation for further calculations, such as determining the stone's volume using its density.

The density of an object is defined as its mass per unit volume. The formula to calculate density is quite straightforward:
\[\[\begin{align*}\text{Density} &= \frac{\text{mass}}{\text{volume}}\end{align*}\]\]In the context of diamonds, this property is especially important. It is what gemologists and jewelers use to identify authentic diamonds, as different materials have different densities.
To find the volume of a diamond, the density formula can be inverted:
\[\[\begin{align*}\text{Volume} &= \frac{\text{mass}}{\text{density}}\end{align*}\]\]For a diamond with a mass of 0.060 grams and a known density of 3.51 \(\text{g}\cdot\text{cm}^{-3}\), the volume calculation becomes a simple task of dividing the mass by density.

Accurately determining the mass and volume of an object like a diamond is a two-step process:
The first step is establishing the mass, which we've converted from carats to grams. Having a precise scale handy is key for measurement, but for homework exercises, the conversion outlined above will get you there.
The second step involves computing the volume using the rearranged density formula. This is crucial if you're working in fields that require precision, such as jewelry design, where knowing the exact volume of a gemstone is necessary for proper setting and mounting.
In the case of our 0.300-carat diamond, first we determined its mass in grams (0.060 g), and then used its density to calculate the volume, leading to our final answer.

Unit conversion is a vital skill in both scientific and everyday contexts. Converting between units enables clear communication and accurate measurement analysis.
For our purposes, we started with a unit of mass specific to gemstones (carats) and converted it to a more universally applicable unit (grams). This also involved understanding the relationship between milligrams and grams.
Being skilled at unit conversion often requires familiarity with several equivalences and sometimes the use of conversion factors. In scientific calculations, consistent units are crucial for obtaining correct results. After converting carats to grams, we were able to apply the known density of diamonds to find the volume of the gemstone in cubic centimeters (cm³).
Honing this ability ensures that mathematical calculations involving different units of measure are accurate and meaningful, whether one is pricing a diamond or conducting scientific research.