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Chapter 1: Problem 70

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

Expert verified
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

01

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: \(V_{cube} = a^3 = (3.00 \ cm)^3 = 27.0 \ cm^3\)
02

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: \(V_{sphere} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (1.42 \ cm)^3 ≈ 12.04 \ cm^3\)
03

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: \(\rho_{cube} = \frac{m_{cube}}{V_{cube}} = \frac{140.4 \ g}{27.0 \ cm^3} ≈ 5.20 \ g/cm^3\)
04

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): \(\rho_{sphere} = \frac{m_{sphere}}{V_{sphere}} = \frac{61.6 \ g}{12.04 \ cm^3} ≈ 5.12 \ g/cm^3\)
05

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.

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Most popular questions from this chapter

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