Question

A package of aluminum foil contains \(50 \mathrm{ft}^{2}\) of foil, which weighs approximately \(8.0\) oz. Aluminum has a density of \(2.70 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the approximate thickness of the foil in millimeters?

Short answer

The approximate thickness of the aluminum foil is \(0.0181 mm\).

Step-by-step solution

Convert given information into appropriate units

First, we need to convert all the values into suitable units for calculation purposes. We have: 1. Area of foil =\(50 ft^2\) 2. Weight of foil = \(8.0 oz\) 3. Density of aluminum = \(2.70 g/cm^3\) We'll convert Area into \(cm^2\), weight into grams (g), and leave density as is, since it is already in SI units. Area of foil: \(1 ft = 30.48 cm\) \(50 ft^2 = 50 × (30.48)^2 cm^2 = 46451.52 cm^2\) Weight of foil: \(1 oz = 28.35 g\) \(8.0 oz = 8.0 × 28.35 g = 226.8 g\) Now we have area = \(46,451.52 cm^2\), weight = \(226.8 g\), and density = \(2.70 g/cm^3\).

Calculate the volume of aluminum foil using mass and density

We can use the following formula to find the volume of aluminum foil: Volume (V) = Mass (m) / Density (ρ) V = \(\frac{226.8}{2.70}\) V = \(84 cm^3\) Now we have the volume as \(84 cm^3\).

Find the thickness of the aluminum foil

Now, we'll calculate the thickness of the aluminum foil by dividing the volume by the area. Thickness (t) = \(\frac{Volume}{Area}\) t = \(\frac{84}{46451.52}\) t = \(0.00181 cm\) Let's convert the thickness into millimeters (mm): \(1 cm = 10 mm\) t = \(0.00181 × 10 = 0.0181 mm\) Hence, the approximate thickness of the aluminum foil is 0.0181 millimeters.

Key concepts

Unit Conversion

Unit Conversion is a fundamental step in solving many scientific and engineering problems. It helps in expressing different measurements in compatible units before calculations.
This process ensures precision and consistency throughout the solution. In the context of the original exercise, we face several unit conversions to make the units consistent and suitable for calculation.

• Area Conversion: Given area of the aluminum foil is 50 square feet (\(50 \mathrm{ft}^{2}\)). To convert it to square centimeters (\(\mathrm{cm}^{2}\)), we use the conversion factor for feet to centimeters. Knowing \(1 \mathrm{ft} = 30.48 \mathrm{cm}\), we compute \(50 \mathrm{ft}^2 = 50 \times (30.48)^2 cm^2 = 46451.52 \mathrm{cm}^2\).


• Weight Conversion: The foil's weight is 8 ounces (oz). We need to convert this into grams. Using the conversion factor \(1 \mathrm{oz} = 28.35 \mathrm{g}\), we calculate \(8 \mathrm{oz} = 8 \times 28.35 \mathrm{g} = 226.8 \mathrm{g}\).


• Density Information: The density is already provided in SI units (\(\mathrm{g/cm}^3\)) and doesn't require conversion.

These conversions lead to more accurate calculations and are crucial for determining properties like volume and thickness correctly.

Volume Calculation

Calculating volume is a crucial step in many physical and chemical processes. In this exercise, we determine the volume of aluminum foil using its mass and density.
Often, volume (\(V\)) can be found using the formula: \[ V = \frac{m}{\rho} \]where \(m\) is mass and \(\rho\) is density.
For the aluminum foil:

• Step Calculation: Given the mass \(m = 226.8 \mathrm{g}\) and density \(\rho = 2.70 \mathrm{g/cm}^3\), the volume becomes \(V = \frac{226.8}{2.70} = 84 \mathrm{cm}^3\).

Most materials with a known density and measurable mass can have their volume calculated this way. It sets the stage for further calculations like finding surface area or thickness.

Thickness Measurement

Determining thickness is vital in material science and various engineering fields. It allows us to understand how thin or thick a material is, which can influence its use and application.
To find thickness, we use the formula:\[ t = \frac{V}{A} \]where \(V\) is the volume and \(A\) is the area.

• From Volume to Thickness: Given volume \(V = 84 \mathrm{cm}^3\) and area \(A = 46,451.52 \mathrm{cm}^2\), the thickness \(t\) is calculated as \(t = \frac{84}{46451.52} = 0.00181 \mathrm{cm}\).


• Final Conversion: Convert thickness from centimeters to millimeters as \(1 \mathrm{cm} = 10 \mathrm{mm}\), resulting in \(t = 0.00181 \times 10 = 0.0181 \mathrm{mm}\).

Understanding how to determine material thickness helps in designing and selecting materials for manufacturing, construction, and packaging.