A solid aluminum sphere has a mass of 85 g. Use the density of aluminum to find the radius of the sphere in inches.

Verified step by step guidance

1. Identify the density of aluminum, which is typically 2.70 g/cm^3.
2. Use the formula for density: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) to find the volume of the sphere. Rearrange to find \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).
3. Substitute the given mass (85 g) and the density of aluminum (2.70 g/cm^3) into the formula to calculate the volume in cm^3.
4. Use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \) to solve for the radius \( r \). Rearrange to find \( r = \left(\frac{3V}{4\pi}\right)^{1/3} \).
5. Convert the radius from centimeters to inches using the conversion factor 1 inch = 2.54 cm.

Key Concepts

Density

Density is defined as mass per unit volume and is a crucial 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 fundamental in determining the size of a sphere based on its volume, which can be derived from its mass and density. Knowing how to manipulate this formula is key to finding the radius from the given mass.

Unit Conversion

Unit conversion is the process of converting a quantity expressed in one set of units to another. In this problem, the radius needs to be expressed in inches, while the calculations may initially yield results in centimeters. Mastery of unit conversion is essential for ensuring that the final answer is in the correct units as specified in the question.