Question

A solid metal sphere has a volume of \(4.2 \mathrm{ft}^{3}\). The mass of the sphere is 155 lb. Find the density of the metal sphere in grams per cubic centimeter.

Short answer

The density of the metal sphere is approximately \(0.5910\ \frac{g}{cm^{3}}\).

Step-by-step solution

Convert volume to cubic centimeters

Given the volume of the sphere is 4.2 ft³. We need to convert this volume to cubic centimeters (cm³). We know that 1 ft = 30.48 cm. Therefore, 1 ft³ = (30.48 cm)³ = 28316.8466 cm³. So, the volume of the sphere in cubic centimeters is: \(4.2\ ft^{3} \times 28316.8466\frac{cm^{3}}{ft^{3}} = 118929.1557\ cm^{3}\)

Convert mass to grams

Given the mass of the sphere is 155 lb. We need to convert this mass to grams (g). We know that 1 lb = 453.59237 g. So, the mass of the sphere in grams is: \(155\ lb \times 453.59237\frac{g}{lb} = 70305.8172\ g\)

Calculate density

We have the volume and mass in the appropriate units (cm³ and g). Now, we can calculate the density using the formula: Density (ρ) = \( \frac{Mass}{Volume} \) \( \rho = \frac{70305.8172 g}{118929.1557 cm^{3}} = 0.5910\ \frac{g}{cm^{3}}\) The density of the metal sphere is approximately 0.5910 grams per cubic centimeter.