A solid aluminum sphere has a mass of 36 g. Use the density of aluminum to find the radius of the sphere in inches.

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Identify the density of aluminum, which is typically 2.70 g/cm³.

Use the formula for density: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) to find the volume of the sphere. Rearrange it to \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).

Substitute the given mass (36 g) and the density of aluminum (2.70 g/cm³) into the volume formula to calculate the volume in cm³.

Use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \) to solve for the radius \( r \). Rearrange it to \( r = \left(\frac{3V}{4\pi}\right)^{1/3} \).

Convert the radius from centimeters to inches using the conversion factor 1 inch = 2.54 cm.

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주요 개념

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Density

Density is defined as mass per unit volume and is a key property of materials. For aluminum, the density is typically around 2.70 g/cm³. Understanding density allows us to relate the mass of an object to its volume, which is essential for solving problems involving geometric shapes like spheres.

Volume of a Sphere

The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius. This formula is crucial for determining the volume of the aluminum sphere based on its mass and density. By rearranging this formula, we can solve for the radius once we have the volume.

Unit Conversion

Unit conversion is the process of converting a quantity expressed in one set of units to another. In this problem, we need to convert the radius from centimeters to inches, as the final answer requires the radius in inches. Knowing the conversion factor (1 inch = 2.54 cm) is essential for accurate results.