The same physical quantity, such as density, can be reported using different units. Above, you found that water has a density of \( 1000 \mathrm{~kg} / \mathrm{m}^{3}=1 \mathrm{~g} / \mathrm{cm}^{3} \). Because the density of water must be the same regardless of what units you use to measure it, you can conclude that an object whose density is \( 1 \mathrm{~kg} / \mathrm{m}^{3} \) must be less dense than water. In other words, \( 1 \mathrm{~kg} / \mathrm{m}^{3} \) is less than \( 1 \mathrm{~g} / \mathrm{cm}^{3} \). If you had three different objects with densities of \( 1 \mathrm{~kg} / \mathrm{m}^{3}, 1 \mathrm{~g} / \mathrm{m}^{3} \), and \( 1 \mathrm{~kg} / \mathrm{mm}^{3} \), which object would be the most dense? Rank the given densities from most to least dense. To rank items as equivalent, overlap them.

To determine which of the three objects is the most dense, let's convert all densities to the same unit for easy comparison. We'll use **kilograms per cubic meter (kg/m³)** as the common unit. 1. **1 kg/m³** - Already in kg/m³. - **Value:** \(1 \, \text{kg/m}^3\) 2. **1 g/m³** - Convert grams to kilograms: \(1 \, \text{g} = 0.001 \, \text{kg}\). - **Value:** \(0.001 \, \text{kg/m}^3\) 3. **1 kg/mm³** - Convert cubic millimeters to cubic meters: \[ 1 \, \text{mm} = 0.001 \, \text{m} \quad \Rightarrow \quad 1 \, \text{mm}^3 = (0.001 \, \text{m})^3 = 1 \times 10^{-9} \, \text{m}^3 \] - Now, convert kg/mm³ to kg/m³: \[ 1 \, \text{kg/mm}^3 = \frac{1 \, \text{kg}}{1 \times 10^{-9} \, \text{m}^3} = 1 \times 10^{9} \, \text{kg/m}^3 \] **Ranking the Densities from Most to Least Dense:** 1. **1 kg/mm³** \(= 1 \times 10^{9} \, \text{kg/m}^3\) 2. **1 kg/m³** \(= 1 \, \text{kg/m}^3\) 3. **1 g/m³** \(= 0.001 \, \text{kg/m}^3\) So, the **most dense** object is the one with a density of **1 kg/mm³**, followed by **1 kg/m³**, and the **least dense** is **1 g/m³**. **Summary:** \[ 1 \, \text{kg/mm}^3 > 1 \, \text{kg/m}^3 > 1 \, \text{g/m}^3 \]