Consider the two spheres shown here, one made of silver and the other of aluminum. (a) What is the mass of each sphere in kg?

검증된 단계별 안내

Determine the volume of each sphere using the formula for the volume of a sphere: \(V = \frac{4}{3} \pi r^3\), where \(r\) is the radius of the sphere.

Identify the density of silver and aluminum from a reliable source. The density of silver is approximately 10.49 g/cm³, and the density of aluminum is approximately 2.70 g/cm³.

Convert the volume of each sphere from cubic centimeters to cubic meters if necessary, as the density is given in g/cm³ and the mass is required in kg.

Calculate the mass of each sphere using the formula: \(\text{mass} = \text{density} \times \text{volume}\). Ensure that the units are consistent, converting grams to kilograms if needed.

Verify the calculations and ensure that the mass is expressed in kilograms for both spheres.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Density

Density is defined as mass per unit volume and is a crucial property of materials. It is typically expressed in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). Knowing the density of a material allows us to calculate its mass if we know its volume, using the formula: mass = density × volume.

Volume of a Sphere

The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. This formula is essential for determining how much space the sphere occupies, which is necessary for calculating its mass when combined with the material's density.

Material Properties

Different materials have distinct properties, including density, which affects their mass for a given volume. Silver and aluminum have different densities (approximately 10.49 g/cm³ for silver and 2.70 g/cm³ for aluminum), which means that even if the spheres are the same size, their masses will differ significantly based on the material they are made from.