Problem 90 - College Algebra with Modeling and Visualization

Question

Thickness of Gold Foil (Refer to Example 9.) A flat, rectangular sheet of gold foil measures 20 centimeters by 30 centimeters and has a mass of 23.16 grams. If 1 cubic centimeter of gold has a mass of 19.3 grams, find the thickness of the gold foil. (Source: U. HaberSchaim, Introductory Physical Science.)

Step-by-Step Solution

Answer

The thickness of the gold foil is 0.002 cm.

Step 1: Determine Volume

First, we need to find the volume of gold foil using the mass provided. The gold foil weighs 23.16 grams. We will use the density of gold, which is 19.3 grams per cubic centimeter, to find the volume. The formula to find volume using mass and density is:\[\text{Volume} = \frac{\text{Mass}}{\text{Density}}\]Substituting the given values, we have:\[\text{Volume} = \frac{23.16 \text{ grams}}{19.3 \text{ grams/cm}^3} = 1.2 \text{ cm}^3\]

Step 2: Calculate Surface Area

Next, calculate the surface area of the gold foil. The dimensions of the gold rectangle are 20 cm by 30 cm. The surface area A is given by the formula for the area of a rectangle:\[A = \text{length} \times \text{width}\]Substituting the given dimensions, we have:\[A = 20 \text{ cm} \times 30 \text{ cm} = 600 \text{ cm}^2\]

Step 3: Calculate Thickness

Now, we use the volume and surface area to calculate the thickness of the foil. The thickness "t" can be found by dividing the volume by the surface area:\[t = \frac{\text{Volume}}{\text{Surface Area}}\]Substituting the calculated values:\[t = \frac{1.2 \text{ cm}^3}{600 \text{ cm}^2} = 0.002 \text{ cm}\]

Step 4: Conclusion

The thickness of the gold foil is calculated as 0.002 centimeters based on the volume and surface area measurements. This shows how a relatively small mass of gold can be spread into a very thin sheet.

Key Concepts

Density

Density is an important concept in science and engineering, as it describes how much mass is contained in a given volume. It is usually expressed in units like grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). To understand it practically, imagine packing a box with marbles. If you put more marbles inside, the box becomes heavier even though its size doesn't change.

The formula for density is:

• Density = Mass / Volume

This tells you how much matter is packed into one unit of volume. In our exercise regarding gold foil, the density was given as 19.3 g/cm³.

This means that each cubic centimeter of gold weighs 19.3 grams. Knowing the density allows us to calculate the volume when we have the mass, as shown in the step-by-step solution. This is especially useful for determining other characteristics, such as thickness, in practical scenarios.

Surface Area

When learning about materials like metal foils, surface area is a key concept. Surface area refers to the total area that the surface of an object occupies. For a rectangular shape, this is easy to calculate using the formula for the area of a rectangle:

• Area = Length × Width

In the problem, we've determined the surface area of a piece of gold foil measuring 20 centimeters by 30 centimeters to be 600 cm².

Understanding surface area helps in many applications, such as determining how thin a material can be spread or how much material is needed to cover a surface. In our example, it helps establish the relationship between the total volume of material and how far it could potentially be spread out, contributing to the calculation of thickness.

Thickness Measurement

The measurement of thickness, especially when dealing with thin materials like foils, is crucial for many industrial and scientific applications. It tells us how thick or thin an object is, generally using units such as millimeters or centimeters.

To find the thickness of a sheet, you need to know both the volume and the surface area of the material. The formula used is quite intuitive:

• Thickness = Volume / Surface Area

In the exercise, the volume of the gold foil was calculated to be 1.2 cm³, and the surface area was 600 cm². This gave us a thickness of 0.002 cm.

Understanding how to measure thickness is useful in various fields, from manufacturing to construction. It allows engineers to ensure that materials meet specifications and are used efficiently, especially when materials like gold need to be optimized for cost and practicality.