Question

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, \(52.7064 \mathrm{~g}\). Calculate the density of the metal to the correct number of significant figures.

Short answer

The density of the metal is 2.6 g/cm³.

Step-by-step solution

Understanding the Density Formula

Density is defined as mass per unit volume. The formula to calculate density is \( \text{density} = \frac{\text{mass}}{\text{volume}} \). To solve this problem, we need to calculate the volume of the rectangular metal bar first and then use the mass to find the density.

Calculate the Volume of the Metal Bar

The volume \( V \) of a rectangular solid can be calculated using the formula: \( V = \text{length} \times \text{width} \times \text{height} \).Substituting the given values, we have:\[ V = 8.53 \text{ cm} \times 2.4 \text{ cm} \times 1.0 \text{ cm} \]\[ V = 20.472 \text{ cm}^3 \].

Determining Significant Figures for Volume

The length, width, and height were measured to 3, 2, and 2 significant figures respectively. Therefore, the volume should be reported to the least number of significant figures, which is 2 (due to width and height). Hence, the volume is \( 20 \text{ cm}^3 \).

Calculate Density

Using the formula for density:\[ \text{density} = \frac{\text{mass}}{\text{volume}} \]We have the mass = 52.7064 g and the volume = 20 cm³.\[ \text{density} = \frac{52.7064 \text{ g}}{20 \text{ cm}^3} \approx 2.63532 \text{ g/cm}^3 \].

Adjust Density to Correct Significant Figures

The measurement with the least number of significant figures in this calculation is the volume (2 significant figures), meaning the density must be reported to 2 significant figures as well:\[ \text{density} = 2.6 \text{ g/cm}^3 \].

Key concepts

Significant Figures

Significant figures are an important concept when it comes to measurement in science and mathematics. They are the digits in a number that carry meaningful information about its measurement precision. Practicing significant figures allows scientists to communicate measurements accurately without overstating precision. Here are some rules about significant figures:

• All non-zero digits are always significant. For example, in 8.53 cm, all three digits are significant.
• Any zeros between significant digits are significant. For instance, in 2.406, all four digits are significant.
• Leading zeros, or zeros before the first non-zero digit, are not significant. For example, 0.004 has one significant figure.
• Trailing zeros in a number with a decimal point are significant. Thus, 2.400 has four significant figures.

For calculations, the number of significant figures in the result is determined by the measurement with the fewest significant figures. This rule is crucial when you perform operations like multiplication or division, as seen when determining density. Using significant figures ensures consistency in accuracy and precision, highlighting the reliability of measurements used in equations.

Volume Measurement

Volume measurement is essential for determining the physical size of an object. In this case, to find the volume of a rectangular metal bar, we use its geometric dimensions. Volume for a rectangular solid is calculated by multiplying its length, width, and height. The equation for this calculation is: \[ V = \text{length} \times \text{width} \times \text{height} \]Given the specific measurements: length = 8.53 cm, width = 2.4 cm, and height = 1.0 cm, the volume calculation will be: \[ V = 8.53 \text{ cm} \times 2.4 \text{ cm} \times 1.0 \text{ cm} \]Upon solving, the volume is 20.472 cm³.It's important to remember that the result of any volume measurement should be reported respecting the number of significant figures. Thus, due to the width and height measurements both having 2 significant figures, our final volume is presented as 20 cm³. Accurate volume measurement and reporting ensure that calculated densities are also precise.

Mass Measurement

Mass measurement is another critical component of density calculation. Mass is a measure of the amount of matter in an object and is typically determined using a balance scale. For this exercise, the mass of the metal bar was given as 52.7064 grams. Here, the mass measurement is very precise, with six significant figures. In mathematical calculations such as finding density, the precision of mass measurement contributes to the overall accuracy of the result. Since density is calculated using the mass and the volume of an object, any inaccuracy in measuring mass will directly affect the calculated density. The principle of significant figures dictates that the result must be aligned with the least precise measurement—in this case, the volume with 2 significant figures. This control ensures that the density calculation remains reliable and appropriately precise.

Rectangular Solid Geometry

Rectangular solid geometry deals with 3-dimensional shapes that have length, width, and height, all perpendicular to each other. Think of a box or a brick; these are everyday examples of rectangular solids. This fundamental geometric concept helps in calculating volumes, which is an integral part of determining the density of materials.In our scenario, the metal bar is a rectangular solid with a length of 8.53 cm, a width of 2.4 cm, and a height of 1.0 cm. Using the volume formula: \[ V = \text{length} \times \text{width} \times \text{height} \]We can easily compute its volume. Understanding rectangular solid geometry allows us to apply simple mathematical principles to find the volume efficiently. Moreover, recognizing these geometric concepts ensures precision in practical situations, like density calculation, which requires careful volume determination as one of its key steps.