Question

A small cube of aluminum measures \(15.6 \mathrm{~mm}\) on a side and weighs \(10.25 \mathrm{~g}\). What is the density of aluminum in \(\mathrm{g} / \mathrm{cm}^{3}\) ?

Short answer

1.75 \text{ g/cm}^{3}

Step-by-step solution

Convert the measurements

First, convert the side length from millimeters to centimeters. Since there are 10 millimeters in a centimeter, divide the side length by 10. So,

Calculate the Volume of the cube

To find the volume of the cube, use the formula for the volume of a cube:

Calculate the Density

To find the density, use the formula:

Key concepts

Density Formula

Density is a measure of how much mass is contained in a given volume. It tells us how tightly matter is packed together in a substance. The formula to calculate density is given by:

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

Where:

• Mass is usually measured in grams (g).
• Volume can be measured in liters (L), cubic centimeters (ich (cm\textsuperscript{3}), or other units of volume.

The unit for density in this exercise is \[ \mathrm{g}/ \mathrm{cm}^{3} \]. To calculate density, you simply divide the mass of the substance by its volume. Using the given values, plug them into the formula to find the density.

Volume Calculation

To calculate the volume of an object, you need to use the appropriate formula. For a cube, the volume is calculated using the formula:

\[ \text{Volume} = \text{side length}^3 \]

Where all sides of the cube are equal. For example, if the side length of the cube is 15.6 mm, you need to ensure that your measurements are in the correct units before performing calculations. Convert side length from millimeters to centimeters by dividing by 10, since there are 10 millimeters in a centimeter. In this example: \[ 15.6 \text{ mm} \div 10 = 1.56 \text{ cm} \]

Now, using the formula: \[ \text{Volume} = (1.56 \text{ cm})^3 \].

Calculate the volume and get the result. This gives you the total volume of the cube in cubic centimeters (\textsuperscript{3}).

Unit Conversion

Unit conversion is crucial when measurements are given in different units. In this exercise, we need to convert the side length from millimeters to centimeters because the density needs to be in \[ \mathrm{g} / \mathrm{cm}^3 \].

Since 1 centimeter (cm) is equal to 10 millimeters (mm), we use the conversion factor: \[ \text{1 cm} = 10 \text{ mm} \]
So, to convert from millimeters to centimeters, you divide the number of millimeters by 10.

For example, a side length of 15.6 mm is converted as follows: \[ 15.6 \text{ mm} \div 10 = 1.56 \text{ cm} \].

Now when calculating volume or density, use these converted units to ensure consistency in your calculations.