Question

A rectangular sheet of aluminum foil measures \(75.0 \mathrm{~cm}\) by \(35.0 \mathrm{~cm}\). What is the thickness of the foil if the volume is \(5.00 \mathrm{~cm}^{3} ?\)

Short answer

The thickness of the foil is approximately \(0.00190 \text{ cm}\).

Step-by-step solution

Understand the problem

We need to calculate the thickness of a rectangular sheet of aluminum foil. We know the length, width, and the volume of the foil. We'll use the formula for volume of a rectangular solid: \( V = ext{length} \times ext{width} \times ext{height} \) where height is the thickness we need to find.

Set up the equation

We know the length \( L = 75.0 \text{ cm} \), width \( W = 35.0 \text{ cm} \), and volume \( V = 5.00 \text{ cm}^3 \). Using the formula, we have: \[ V = L \times W \times h \]which translates to: \[ 5.00 \text{ cm}^3 = 75.0 \text{ cm} \times 35.0 \text{ cm} \times h \]

Solve for thickness

Rearrange the equation to solve for the thickness \( h \): \[ h = \frac{V}{L \times W} \]Substituting the known values, we get: \[ h = \frac{5.00 \text{ cm}^3}{75.0 \text{ cm} \times 35.0 \text{ cm}} \] Calculate this to find \( h \).

Calculate the result

Perform the calculation: \[ h = \frac{5.00}{75.0 \times 35.0} \]This simplifies to: \[ h = \frac{5.00}{2625.0} \]Now, divide to find \( h \): \[ h = 0.00190476 \text{ cm} \] Thus, the thickness of the foil is approximately \(0.00190 \text{ cm}\) when rounded to four significant figures.

Key concepts

Volume Calculations

Volume calculations are fundamental in geometry and help us determine how much space an object occupies. When dealing with three-dimensional objects like a rectangular solid, the formula for volume takes into account the object's length, width, and height. The volume is calculated using the formula:\[ V = \text{length} \times \text{width} \times \text{height} \]Each element in the formula corresponds to a dimension of the object:

• Length: This is how long the object is from one end to the other.
• Width: This is the side-to-side measurement across the object's base.
• Height: This is how tall the object is, or in some contexts, the thickness.

In the context of the exercise, the volume given was the total space that the foil occupies despite it being mostly flat. By knowing the volume and two of the dimensions, we can solve for the unknown dimension, which in this case was the thickness of the foil.

Rectangular Solid

A rectangular solid is one of the most common geometric shapes, resembling a box or a brick. It is defined by three dimensions: length, width, and height. This shape is also referred to as a rectangular prism and can be found in everyday objects, like a shoebox.Understanding the characteristics of a rectangular solid is essential when solving problems related to its volume:

• All angles: Each angle in a rectangular solid is a right angle (90 degrees).
• Parallel Faces: Opposite faces are equal and parallel.
• Edges: Opposite edges are of equal length.

The formula for calculating the volume of a rectangular solid, \( V = \text{length} \times \text{width} \times \text{height} \), captures how this shape expands across three dimensions, giving it its three-dimensional form.

Mathematical Problem Solving

Mathematical problem solving is an essential skill that involves understanding a problem, devising a plan, carrying out the plan, and reviewing the solution. Within this framework, we tackle problems systematically and logically. The exercise involving the aluminum foil is a perfect example of this process. Here's how problem solving is applied in this exercise:

• Understand the Problem: We first identify what's known and what needs to be discovered—in this case, the thickness of the foil.
• Set Up the Equation: By interpreting the given information, we establish an equation using the known volume formula.
• Solve for the Unknown: The equation is rearranged to isolate the unknown variable, allowing us to compute its value.
• Check the Solution: Verification ensures that the calculations are correct and the solution is sensible given the context of the problem.

When solving mathematical problems, always ensure that each step builds on the previous one, maintaining coherence through logical reasoning and accurate calculations.