Question

A sheet of aluminum (Al) foil has a total area of \(1.000 \mathrm{ft}^{2}\) and a mass of \(3.636 \mathrm{~g}\). What is the thickness of the foil in millimeters (density of \(\left.\mathrm{Al}=2.699 \mathrm{~g} / \mathrm{cm}^{3}\right)\) ?

Short answer

The thickness of the aluminum foil is 0.0145 mm.

Step-by-step solution

Convert the Area from Square Feet to Square Centimeters

First, we need to convert the given area from square feet to square centimeters. We know that 1 square foot = 929.0304 square centimeters. Therefore, 1.000 square feet = 929.0304 square centimeters.

Use Density to Find Volume

Next, use the density formula to find the volume of the aluminum foil. The density formula is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}}. \]Rearranging it gives:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}}. \]The mass is 3.636 g and the density is 2.699 g/cm³. Thus, the volume \( V \) is:\[ V = \frac{3.636 \text{ g}}{2.699 \text{ g/cm}^3} \approx 1.347 \text{ cm}^3. \]

Calculate the Thickness in Centimeters

The aluminum foil is a thin flat object, so its volume can also be expressed as:\[ \text{Volume} = \text{Area} \times \text{Thickness}. \]We rearrange this to find thickness:\[ \text{Thickness} = \frac{\text{Volume}}{\text{Area}}. \]Using the calculated volume and converted area, we have:\[ \text{Thickness} = \frac{1.347 \text{ cm}^3}{929.0304 \text{ cm}^2} \approx 0.00145 \text{ cm}. \]

Convert Thickness to Millimeters

Finally, convert the thickness from centimeters to millimeters, knowing that 1 cm = 10 mm:\[ \text{Thickness} = 0.00145 \text{ cm} \times 10 \frac{\text{mm}}{\text{cm}} = 0.0145 \text{ mm}. \]

Key concepts

Unit Conversion

To solve problems involving different measurement units, mastering unit conversion is essential. Understanding that different unit systems are used in various contexts helps in making accurate calculations. For example, square feet and square centimeters are units for measuring area. Key Conversion Facts

• 1 square foot = 929.0304 square centimeters
• 1 centimeter = 10 millimeters

When converting, it's crucial to multiply by the right conversion factor. For instance, if you are given area in square feet and need it in square centimeters, multiply by 929.0304. If converting length from centimeters to millimeters, multiply by 10. These straightforward multiplication processes are vital for precise unit conversions.

Volume Calculation

Volume calculation is a fundamental step when you have a given mass and need to find dimensions such as thickness. This is especially true when working with objects like sheets or foils.The Density Formula and Rearrangement
The formula for density connects mass and volume and is written as:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}}. \]To solve for volume, rearrange the formula as:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}}. \]This simple rearrangement allows you to find the volume from known quantities of mass and density. For example, with the aluminum foil, given a mass of 3.636 g and density of 2.699 g/cm³, we calculated the volume as approximately 1.347 cm³.This step is crucial for determining dimensions like thickness from given material properties.

Thickness Measurement

Thickness measurement comes in handy when dealing with thin materials like aluminum foil.To find thickness using volume, you can use the formula:Volume Relationship
For thin objects, you can express volume as:\[ \text{Volume} = \text{Area} \times \text{Thickness}. \]To solve for thickness, rearrange as:\[ \text{Thickness} = \frac{\text{Volume}}{\text{Area}}. \]This expression helps you relate the three dimensions to find thickness if area and volume are known. In the exercise, with a volume of approximately 1.347 cm³ and an area of 929.0304 cm², the calculated thickness was about 0.00145 cm. This relationship aids in measuring tiny dimensions of materials effectively.

Aluminum Properties

Aluminum is a popular metal known for its useful properties and applications. Understanding its properties is essential for applications in industries like packaging, automotive, and aerospace. General Properties

• Lightweight yet strong
• Density: 2.699 g/cm³
• Corrosion-resistant

Due to its relatively low density compared to other metals, aluminum is ideal for making sheets and foils. Its properties allow for easy handling and transformation into various forms, such as the thin aluminum foil discussed here. Knowing these characteristics helps inform the choice of aluminum for different uses.