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 \(0.591 \mathrm{g/cm^3}\).

Step-by-step solution

Convert the volume from cubic feet to cubic centimeters

To convert the volume from cubic feet to cubic centimeters, we can use the following conversion factor: 1 ft³ = 28,316.8 cm³ Given volume = 4.2 ft³ So, to convert the given volume to cubic centimeters, we multiply by the conversion factor: Volume = 4.2 ft³ * 28,316.8 cm³/ft³

Calculate the volume in cubic centimeters

Now, we can plug in the numbers to find the volume in cubic centimeters: Volume = 4.2 * 28,316.8 cm³/ft³ ≈ 118,930.56 cm³

Convert the mass from pounds to grams

The given mass is in pounds. We need to convert it to grams using the following conversion factor: 1 lb = 453.592 g Given mass = 155 lb So, to convert the given mass to grams, we multiply by the conversion factor: Mass = 155 lb * 453.592 g/lb

Calculate the mass in grams

Now, we can plug in the numbers to find the mass in grams: Mass = 155 * 453.592 g/lb ≈ 70,306.24 g

Calculate the density using the formula

The formula for density is: Density = Mass / Volume Now, we can plug in the values we found for the mass and volume: Density = 70,306.24 g / 118,930.56 cm³

Calculate the density of the metal sphere

Now, we can compute the final answer: Density ≈ 0.591 g/cm³ The density of the metal sphere is approximately 0.591 grams per cubic centimeter.

Key concepts

Unit Conversion

Understanding unit conversion is essential in scientific calculations where measurements may be given in a variety of units. For the density calculation, we often need to convert mass and volume units to a standard form in order to compute the result in a desired unit.

Example of Volume Conversion:
When the exercise provides the volume of an object in cubic feet (ft³) and we need to find the density in grams per cubic centimeter (g/cm³), a conversion is necessary. One cubic foot equals 28,316.8 cubic centimeters. Thus, by multiplying the volume in cubic feet by this conversion factor, we obtain the volume in cubic centimeters.

Example of Mass Conversion:
Similarly, if the mass is given in pounds (lb), and the desired density unit is grams per cubic centimeter, we convert the mass from pounds to grams by using the conversion factor where one pound equals 453.592 grams.

Mass-Volume Relationship

In the context of density, the mass-volume relationship is pivotal. This relationship indicates how much mass is contained in a given volume. A common challenge when dealing with this kind of problem is ensuring that the mass and volume are in compatible units before calculating the density.

To correctly utilize the mass-volume ratio, accurate measurement or conversion of mass and volume to coherent units is critical. Only then can we proceed to determine the density. For instance, if we have the volume of an object in cubic centimeters and the mass in grams, we can directly apply these values to the density formula. The mass-volume relationship is what allows us to understand the underlying concept of density as a measure of how tightly matter is packed within a substance.

Density Formula

The density formula is fundamental for calculating how dense a substance is. Density is defined as the mass of an object divided by its volume. Mathematically, it is expressed as:

\[ Density = \frac{Mass}{Volume} \]

This formula allows you to determine the density once you have the mass and volume of an object. Remember to ensure that mass and volume are given in consistent units for the density to be accurate. In our metal sphere example, once the mass is converted to grams and the volume to cubic centimeters, these values are inserted into the density formula to find that the metal's density is approximately 0.591 g/cm³. The resulting value tells us how much gram-weight of the substance is present per unit volume (cubic centimeter) of the material.