Question

The density of gold is 19.3 \(\mathrm{g} / \mathrm{cm}^{3}\) . a. What is the volume, in cubic centimeters, of a sample of gold that has a mass of 0.715 \(\mathrm{kg}\) ? b. If this sample of gold is a cube, what is the length of each edge in centimeters?

Short answer

The volume is approximately 37.05 \text{cm}^3\ and the edge length of the cube is approximately 3.34 \text{cm}\.

Step-by-step solution

- Convert mass to grams

The mass of the gold sample is given in kilograms. Convert it to grams since the density is in \(\text{g/cm}^3\). \(0.715 \text{kg} = 715 \text{g}\)

- Use the density formula

Use the formula for density: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\). Rearrange it to solve for volume: \(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\). Substitute the values: \( \text{Volume} = \frac{715 \text{g}}{19.3 \text{g/cm}^3} \).

- Calculate the volume

Perform the division to find the volume: \( \text{Volume} = \frac{715}{19.3} \approx 37.05 \text{cm}^3 \).

- Volume of the cube

For part b, if the sample is a cube, its volume is \( V = \text{side}^3 \). So, \( \text{side} = \sqrt[3]{\text{Volume}}\).

- Calculate the edge length of the cube

We know that \( \text{Volume} \approx 37.05 \text{cm}^3\), so calculate \( \text{side} = \sqrt[3]{37.05} \approx 3.34 \text{cm}\).

Key concepts

density formula

Understanding the density formula is key to solving problems involving mass and volume. The density formula expresses the relationship between mass, volume, and density. It is typically written as:

\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
By rearranging this formula, you can solve for any of the three variables, provided you have the other two. This flexibility makes the density formula a valuable tool in various scientific calculations. For example, to find the volume when you know the mass and density, you rearrange it to:

\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
This principle underpins many practical applications, such as determining the space a substance will occupy or figuring out how heavy an object of a certain size will be.

mass to volume conversion

When solving problems related to density, converting mass to volume (or vice versa) is a crucial step. The exercise exemplifies this with the given mass of gold. Since density is often provided in units of grams per cubic centimeter (\(\text{g/cm}^3\)), you might need to convert the mass to grams if it's given in kilograms.

For instance, 0.715 kilograms is converted to 715 grams to match the units of density. Once the mass is in the correct units, you can use the density formula to find the volume. In this case, we use:

\[ \text{Volume} = \frac{715 \text{g}}{19.3 \text{g/cm}^3} \rightarrow \text{Volume} \text{ (V)} \text{ is approximately } 37.05 \text{ cm}^3 \]
Remember, keeping the units consistent is vital in ensuring accurate calculations.

cubic measurements

Cubic measurements are essential when dealing with three-dimensional objects. In the exercise, once we know the volume of the gold sample, we assume it is a cube to find the length of its edges. A cube's volume is given by:

\[ \text{Volume (V)} = \text{side}^3 \]
We can use the calculated volume of about 37.05 cm^3 to find the length of each edge. By taking the cube root of the volume, we determine the side length:

\[ \text{side} = \root{3}{37.05} \right\text{ side is approximately } 3.34 \text{ cm} \]
This means that each edge of the cube-shaped gold sample will be around 3.34 centimeters long. Cubic measurements are hence critical for understanding and solving real-world problems involving volume and dimensions of objects.

mathematical problem solving

Mathematical problem-solving skills are invaluable for navigating exercises like the one provided. These skills involve:

• Identifying the known variables and the unknowns
• Choosing the correct formulas
• Making the necessary unit conversions
• Performing calculations accurately

For instance, in the gold density problem, we first convert the mass from kilograms to grams. Next, we apply the density formula to solve for volume. Finally, knowing the volume, we compute the edge length of the cube. These problem-solving steps illustrate how to methodically approach and solve mathematical exercises, ensuring you get the correct solutions by breaking down the problem into manageable parts. Clear thinking and organized calculations can simplify even complex problems.