Answer
$$48.7\%$$
Work Step by Step
1. Write the system of equations:
$$Total \space mass = 4.36 \space kg \times \frac{1000 \space g}{1 \space kg} = 4360 \space g = m_c + m_l$$ $$Total \space volume = 427 \space cm^3 = v_c + v_l$$ $$Density \space (copper) = 8.96 \space g/cm^3 = \frac{m_c}{v_c}$$ $$Density \space (lead) = 11.4 \space g/cm^3 = \frac{m_l}{v_l}$$
2. Solve for $v_c$ and $v_l$ in the density equations, and substitute their values into the volume equation. $$ 8.96 \space = \frac{m_c}{v_c} \longrightarrow v_c = \frac{m_c}{8.96} = \frac 1 {8.96}m_c $$ $$ 11.4 \space = \frac{m_l}{v_l} \longrightarrow v_l = \frac{1}{11.4} m_l$$ $$427 = v_c + v_l = \frac 1{8.96}m_c + \frac 1 {11.4}m_l \approx 0.1116m_c + 0.08772 m_l$$
3. Solve for $m_c$ in the first equation, and substitute its value into the last equation that we wrote. $$m_c = 4360 - m_l$$ $$427 = 0.1116(4360 - m_l) + 0.08772m_l$$ $$427 = 486.6 - 0.1116m_l + 0.08772m_l$$ $$-59.6 = -0.02388 m_l$$ $$m_l = \frac{-59.6}{-0.02388} = 2495.8 \space g$$
4. Find the other values.
$$m_c = 4360 - m_l = 4360 - 2495.8 = 1864.2$$ $$v_l = \frac 1 {11.4} m_l = 219 $$ $$v_c = \frac{1} {8.96} m_c = 208$$
5. Calculate the percentage of copper spheres:
- If all spheres have the same volume, the percent volume of copper is equal to the percentage of copper spheres:
$$\frac{208}{427}\times 100\% = 48.7\%$$
