Let's break down the solution step-by-step:
1. Identify the given quantities:
- Density of material A: [tex]\( \rho_A = 2.5 \, \text{g/cm}^3 \)[/tex]
- Density of material B: [tex]\( \rho_B = 3.0 \, \text{g/cm}^3 \)[/tex]
- Mass of material A: [tex]\( m_A = 400 \, \text{g} \)[/tex]
- Mass of material B: [tex]\( m_B = 480 \, \text{g} \)[/tex]
2. Calculate the volumes of materials A and B:
- Volume is calculated using the formula [tex]\( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \)[/tex].
- For material A:
[tex]\[
V_A = \frac{m_A}{\rho_A} = \frac{400 \, \text{g}}{2.5 \, \text{g/cm}^3} = 160 \, \text{cm}^3
\][/tex]
- For material B:
[tex]\[
V_B = \frac{m_B}{\rho_B} = \frac{480 \, \text{g}}{3.0 \, \text{g/cm}^3} = 160 \, \text{cm}^3
\][/tex]
3. Calculate the total mass and total volume of the mixture:
- Total mass:
[tex]\[
m_{\text{total}} = m_A + m_B = 400 \, \text{g} + 480 \, \text{g} = 880 \, \text{g}
\][/tex]
- Total volume:
[tex]\[
V_{\text{total}} = V_A + V_B = 160 \, \text{cm}^3 + 160 \, \text{cm}^3 = 320 \, \text{cm}^3
\][/tex]
4. Calculate the density of the mixture:
- Density of the mixture is given by:
[tex]\[
\rho_{\text{mixture}} = \frac{m_{\text{total}}}{V_{\text{total}}} = \frac{880 \, \text{g}}{320 \, \text{cm}^3} = 2.75 \, \text{g/cm}^3
\][/tex]
Thus, the density of the mixture is [tex]\( 2.75 \, \text{g/cm}^3 \)[/tex].
