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 \(0.01810\) millimeters.

Step-by-step solution

Find the weight of the aluminum foil in grams

Since the weight of the foil is given in ounces, we need to convert it to grams. We can use the conversion factor 1 oz ≈ 28.3495 g: Weight (g) = 8.0 oz * 28.3495 g/oz ≈ 226.796 g

Find the volume of the aluminum foil

We can find the volume of the aluminum foil by using its density (mass per unit volume) and the weight of the foil. The formula to find the volume is: Volume = Weight/Density The weight of the foil is 226.796 g, and the density of aluminum is 2.70 g/cm³. So, we have: Volume = 226.796 g / 2.70 g/cm³ ≈ 84.036 cm³

Convert the surface area of the foil to square centimeters

The surface area of the foil is given in square feet. To calculate the thickness, we need the surface area in square centimeters. We can use the conversion factor 1 ft² ≈ 929.03 cm²: Surface area (cm²) = 50 ft² * 929.03 cm²/ft² ≈ 46451.5 cm²

Find the thickness of the aluminum foil

We can find the thickness of the foil by dividing the volume of the foil by its surface area. The formula to find the thickness is: Thickness = Volume / Surface area The volume of the foil is 84.036 cm³, and the surface area is 46451.5 cm². So, we have: Thickness = 84.036 cm³ / 46451.5 cm² ≈ 0.001810 cm

Convert the thickness to millimeters

Finally, we need to convert the thickness from centimeters to millimeters. We can use the conversion factor 1 cm = 10 mm: Thickness (mm) = 0.001810 cm * 10 mm/cm ≈ 0.01810 mm The approximate thickness of the aluminum foil is 0.01810 millimeters.

Key concepts

Density

When we discuss the concept of density, we are talking about how much mass is contained within a certain volume. It's a fundamental property of materials and is expressed as mass per unit volume, typically in grams per cubic centimeter (g/cm³) for solids. In the context of aluminum foil, density is used to determine its volume given a specific weight.

Imagine filling a tiny 1 cm x 1 cm x 1 cm cube with aluminum. The density of aluminum, 2.70 g/cm³, informs us that this cube would weigh 2.70 grams. This relates back to the exercise where we know the weight of the aluminum foil and need to figure out how much space that weight occupies (the volume). Density is like a magic formula that helps bridge the gap between weight and volume in order to eventually find out the thickness of the foil.

Unit Conversion

Unit conversion is a crucial process in solving a variety of scientific and technical problems when you must transition between different units of measurement. Whether you're dealing with length, mass, volume, or area, mastering unit conversions allows seamless comparisons and calculations.

Let's take our aluminum foil example; the weight starts in ounces but to work with density, it needs to be in grams. Similarity arises with the surface area initially provided in square feet needing conversion to square centimeters. Without proper unit conversions, it would be impossible to progress with our calculations. Unit conversion hinges on the use of constants like 1 oz is approximately 28.3495 g or 1 ft² is 929.03 cm², making it possible to navigate between different measurement systems easily.

Volume Calculation

Calculating volume is about determining how much three-dimensional space an object occupies. In mathematical terms, volume is found by multiplying the area of the base by the height of the object. But for substances where you can't directly measure dimensions, you rely on formulas that incorporate density and mass.

In our exercise, the volume of aluminum foil isn't a straightforward measurement—it's derived from its known mass and density. By rearranging the density formula, we get Volume = Mass / Density, which allows us to calculate the volume of irregular materials or objects which, like aluminum foil, don't have easily measured dimensions. The calculated volume is the crux in finding out the final thickness of the foil.

Surface Area

Surface area is essentially how much exposed area an object has. In most simple shapes, like cubes and spheres, there are specific formulas to calculate surface area. In the case of aluminum foil, we're dealing with a flat sheet, and surface area is notably straightforward—it's the length times the width.

However, the challenge arises when the units aren't compatible with other measurements in a problem, as we saw where the area was given in square feet but needed to be in square centimeters. To find the thickness of the foil, we need to understand its surface area accurately, as it is one of the two key pieces of information needed to complete our calculation. Once converted to the correct unit, you divide the volume by the surface area to unravel the mystery of the foil's thickness.