In determining the density of a rectangular metal bar, a student made the following measurements: length, \(8.53 \mathrm{~cm} ;\) width, \(2.4 \mathrm{~cm} ;\) height, \(1.0 \mathrm{~cm}\) mass, 52.7064 g. Calculate the density of the metal to the correct number of significant figures.

Understanding significant figures is crucial when reporting the results of a calculation in science and engineering. These figures are the digits in a number that carry meaning contributing to its precision. This typically includes all digits except:

• Leading zeros (zeros before non-zero digits).
• Trailing zeros in a number without a decimal point.
• Spurious digits obtained from calculations exceeding the precision of the raw data.

For instance, in the measurement of the metal bar length (8.53 cm), all three numbers are significant because they are non-zero and provide precision to the measurement. The exactness of the final answer must reflect the least precise measurement used in the calculations to maintain the integrity of the data. This concept guides the rounding of the metal bar's density to two significant figures, following the lead of the height measurement (1.0 cm), which only has two.

The calculation of volume for geometric shapes follows specific formulas based on their geometry. For prisms, such as the rectangular metal bar in our exercise, volume is determined by the product of its length, width, and height. Mathematically, this is expressed as Volume = Length Width Height.

Calculating the volume correctly is imperative as it impacts subsequent calculations, such as density. In our example, the volume is calculated by multiplying the values (8.53 cm 2.4 cm 1.0 cm), resulting in 20.472 cm^3. It's essential to pay attention to the precision of measurements and to maintain a consistent unit of measurement to ensure accuracy.

Standardizing the units of measurement in any calculation is fundamental to obtain a correct and meaningful result. In physics and chemistry, the International System of Units (SI) is used, with meters (m) as the base unit for measuring length, volume in cubic meters (m^3), and mass in kilograms (kg).

In our exercise, we work within the centimeter-gram-second (CGS) system, using centimeters (cm) to measure dimensions and grams (g) for mass. Therefore, the volume comes out in cubic centimeters (cm^3), which matches with the mass's unit grams to calculate the density in grams per cubic centimeter (g/cm^3). Consistency in units ensures that calculations like density are dimensionally accurate.

The mass-to-volume ratio, commonly known as density, is a property of materials that describes the amount of mass per unit volume and is expressed using the formula:
Density = Mass / Volume
This ratio is crucial in identifying substances and understanding their properties. A higher density means that a substance contains more mass in a given volume than a substance with a lower density.

In the example of the metal bar, its density was originally calculated as 2.574 g/cm^3 but was later rounded to 2.6 g/cm^3 to conform with the significant figures represented by the least precise measurement. Accurately determining the mass-to-volume ratio involves not only precise measurements but also careful consideration of significant figures, ensuring the ratio accurately reflects the object's physical characteristics.