A box contains a mixture of small copper spheres and small lead spheres. The total volume of both metals is measured by the displacement of water to be 427 cm³, and the total mass is 4.36 kg. What is the percentage of copper spheres?

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Identify the densities of copper and lead. The density of copper is approximately 8.96 g/cm³, and the density of lead is approximately 11.34 g/cm³.

Let the volume of copper be V_c and the volume of lead be V_l. We know that V_c + V_l = 427 cm³.

Let the mass of copper be m_c and the mass of lead be m_l. We know that m_c + m_l = 4360 g (since 4.36 kg = 4360 g).

Using the density formula, \( \text{density} = \frac{\text{mass}}{\text{volume}} \), express the mass of copper as m_c = 8.96 \times V_c and the mass of lead as m_l = 11.34 \times V_l.

Substitute the expressions for m_c and m_l into the mass equation: 8.96 \times V_c + 11.34 \times V_l = 4360. Solve the system of equations to find V_c and V_l, then calculate the percentage of copper as \( \frac{V_c}{427} \times 100 \).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density

Density is defined as mass per unit volume and is a critical property of materials. It is calculated using the formula density = mass/volume. Understanding density allows us to relate the mass and volume of the copper and lead spheres in the mixture, which is essential for determining the percentage of each metal based on their respective densities.

Volume Displacement

Volume displacement is a method used to measure the volume of an irregular object by submerging it in a fluid, typically water. The volume of water displaced is equal to the volume of the object. In this question, the total volume of the copper and lead spheres is determined through water displacement, which provides the necessary information to analyze the mixture.

Mass Percentage

Mass percentage is a way to express the concentration of a component in a mixture, calculated as (mass of component/total mass) × 100%. In this scenario, calculating the mass percentage of copper spheres requires determining the mass of copper in relation to the total mass of the mixture, allowing for a clear understanding of the composition of the metal spheres.