Problem 58 - Algebra 1: Concepts and Skills

Question

You are making candles. You melt paraffin wax in the cubic container shown below. Each edge of the container is 6 inches in length. The container is half full. What is the volume of the wax in this container?

Step-by-Step Solution

Answer

The volume of the wax in the container is 108 cubic inches.

Step 1: Calculation of the volume of the cubic container

In 3D Geometry, the formula for the volume of a cube is the cube of edge length. Given, the edge length of the cubic container is 6 inches, so the volume would be \(6^3 = 216\) cubic inches.

Step 2: Calculate the volume of wax in the container

The container is half full, so the volume of the wax would be half the volume of the cubic container. Hence, it is \(216 / 2 = 108\) cubic inches.

Key Concepts

Understanding 3D Geometry

Three-dimensional (3D) geometry includes shapes that have three dimensions: length, width, and height. Unlike two-dimensional shapes, which lie flat on a plane, 3D shapes like cubes, spheres, and cylinders have depth. This depth is what gives 3D objects their volume, a measure of how much space they occupy inside.
Understanding 3D geometry helps us to visualize and calculate the volume or space within an object. For example, in 3D geometry, a cube has all sides equal, making it a special type of rectangular prism. Knowing how to work with 3D shapes, especially cubes, is essential in various real-life applications, including storage and packaging, as well as fun projects like candle making.

Exploring Paraffin Wax

Paraffin wax is a type of wax made from petroleum, coal, or oil shale. It's commonly used in candle making due to its excellent burning properties. It melts easily and holds scent and color well, making it a popular choice for handmade candles.
In the context of the exercise, paraffin wax serves as the material being melted and poured into the cubic container. Understanding the properties of paraffin wax—like its melting point and how it behaves when solid or liquid—can help in ensuring the candle-making process is smooth. It's important to know the volume of liquid paraffin so you can fill the container accurately without overflowing.

Features of a Cubic Container

A cubic container is a 3D shape with all six sides being equal squares. This means each edge of the container is the same length, which greatly simplifies calculations related to its volume.
For practical uses, a cubic container's features make it easy to evenly distribute and store substances like wax. In this exercise, the cubic container has a side length of 6 inches, offering a clear example of how measured volume is accessible. By knowing one simple measurement—its side length—you can determine all other volumetric details needed.

Calculation of Volume Made Simple

The volume of a 3D object is a measure of how much space it takes up. Calculating the volume of a cube is straightforward because all of its sides are equal.
To find the volume, you use the formula for the volume of a cube: \[\text{Volume} = \text{side length}^3\]In this problem, the side of the cubic container is 6 inches. Thus, the volume is \[6^3 = 216 \text{ cubic inches}\].
To find the volume of the paraffin wax, remember the container is only half full. Simply divide the total volume by 2, resulting in \[216 / 2 = 108 \text{ cubic inches}\]. Here you can see the step-by-step process not only simplifies a typical calculation, but also applies practical math skills in everyday situations like candle crafting.