Question

One metal object is a cube with edges of 3.00 \(\mathrm{cm}\) and a mass of 140.4 \(\mathrm{g} .\) A second metal object is a sphere with a radius of 1.42 \(\mathrm{cm}\) and a mass of 61.6 \(\mathrm{g} .\) Are these objects made of the same or different metals? Assume the calculated densities are accurate to \(\pm 1.00 \%\) .

Short answer

The cube has a density of approximately \(5.20 \ g/cm^3\) and the sphere has a density of approximately \(5.12 \ g/cm^3\). The percentage difference in density between the two objects is approximately \(1.54\%\) which is greater than the allowed ±1.00%, so the two objects are made of different metals.

Step-by-step solution

Calculate the volume of the cube

To find the density, we need the volume of the cube. The formula to calculate the volume of a cube is V = a³, where a is the edge length of the cube. In this case, a = 3.00 cm. Calculate the volume: \(V_{cube} = a^3 = (3.00 \ cm)^3 = 27.0 \ cm^3\)

Calculate the volume of the sphere

The formula to find the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere. In this case, the radius r = 1.42 cm. Calculate the volume: \(V_{sphere} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (1.42 \ cm)^3 ≈ 12.04 \ cm^3\)

Calculate the density of the cube

Density (ρ) is calculated using the formula ρ = mass/volume. The mass of the cube is given as 140.4 g, and we have already calculated its volume (27.0 cm³). Now we can calculate the density of the cube: \(\rho_{cube} = \frac{m_{cube}}{V_{cube}} = \frac{140.4 \ g}{27.0 \ cm^3} ≈ 5.20 \ g/cm^3\)

Calculate the density of the sphere

Similarly, we can calculate the density of the sphere using its mass (61.6 g) and its volume (12.04 cm³): \(\rho_{sphere} = \frac{m_{sphere}}{V_{sphere}} = \frac{61.6 \ g}{12.04 \ cm^3} ≈ 5.12 \ g/cm^3\)

Calculate the percentage difference in density

To determine whether the objects are made of the same or different metals, we need to compare their densities. We can calculate the percentage difference between the densities: \(\% \ difference = \frac{|\rho_{cube} - \rho_{sphere}|}{(\frac{\rho_{cube} + \rho_{sphere}}{2})} \times 100\) \(\% \ difference = \frac{|5.20 - 5.12|}{(\frac{5.20 + 5.12}{2})} \times 100 ≈ 1.54\%\) Since the percentage difference in density is 1.54%, which is greater than ±1.00%, we can conclude that the objects are made of different metals.

Key concepts

Volume of Cube

The volume of a cube is essential when calculating its density, especially when you want to understand the material it is made from. A cube is a 3D shape with six equal square faces, which makes calculating its volume straightforward. If you have a cube, the formula to find its volume is given by the formula: \[V = a^3\]where \(a\) represents the length of one edge of the cube. For example, if each edge of a cube is 3.00 cm long, then its volume would be:

• \(V = (3.00 \, \text{cm})^3 = 27.0 \, \text{cm}^3\)

This calculation provides the space enclosed within the cube, which can then be used in further computations, such as determining the cube's density. Calculating volume accurately is crucial as it serves as a prerequisite step for many other physics and engineering tasks.

Volume of Sphere

To find the volume of a sphere, which is important for evaluating its density, you use a slightly more complex method than you do for a cube. A sphere is a perfectly round 3D shape, and its volume can be calculated with the formula:\[V = \frac{4}{3} \pi r^3\]where \(r\) is the radius of the sphere. For instance, if a sphere has a radius of 1.42 cm, its volume can be calculated as:

• \(V = \frac{4}{3} \pi (1.42 \, \text{cm})^3 \approx 12.04 \, \text{cm}^3\)

This volume is a measure of how much space the sphere occupies. Understanding how to calculate the volume of a sphere is useful, especially in fields like physics and materials science, where spheres often represent atoms or other small particles.

Density Formula

Density is a fundamental concept that helps in determining the composition of materials. It is defined as mass per unit volume and is represented by the formula:\[\rho = \frac{m}{V}\]where \(\rho\) is the density, \(m\) is the mass, and \(V\) is the volume.
To find the density of a cube, for example, you would require its mass and previously calculated volume:

• If the cube has a mass of 140.4 g and a volume of 27.0 cm³, its density would be \( \rho = \frac{140.4 \, \text{g}}{27.0 \, \text{cm}^3} \approx 5.20 \, \text{g/cm}^3\).

Similarly, for a sphere with a mass of 61.6 g and volume of 12.04 cm³, its density would be calculated as:

• \( \rho = \frac{61.6 \, \text{g}}{12.04 \, \text{cm}^3} \approx 5.12 \, \text{g/cm}^3\).

Density helps in materials analysis, aiding in the identification of substances based on their mass-volume ratio.

Metal Composition Analysis

Analyzing the composition of metals involves understanding their densities and comparing them. Such analysis can determine whether two objects are made from the same material or not. Once densities are calculated, the next step is to look at their percentage difference to decide if the materials match.
In the example given, the densities calculated are:

• Cube: 5.20 g/cm³
• Sphere: 5.12 g/cm³

The percentage difference in their densities provides insight into their metal composition:

• The formula is: \[\% \text{difference} = \frac{|\rho_{cube} - \rho_{sphere}|}{\frac{(\rho_{cube} + \rho_{sphere})}{2}} \times 100\]

With this approach, the percentage difference is found to be approximately 1.54%, which suggests that the objects are not composed of the same metal, given their percentage difference exceeds the assumed accuracy range of ±1.00%. This process is crucial in materials science and engineering, as it helps identify unknown metals and verify product specifications.