Problem 59 - 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. Each edge of a second cubic container is 4 inches in length. Can this second candle mold hold the same amount of melted wax that is in the candle mold shown at the right? Explain your answer.

Step-by-Step Solution

Answer

After calculation, we find that the volume of the wax in the first cube is \(0.5 * 6^3 = 108\) cubic inches, and the volume of wax that the second cube can hold is \(4^3 = 64\) cubic inches. Therefore, the second cube cannot hold the same amount of wax as the first cube.

Step 1: Compute the Volume of Wax in the First Cube

We start by calculating the volume of the wax in the first cube. The side length of the first cube is given as 6 inches. The cube is half full, so the volume of the wax is half the volume of the cube. To find the volume, we cube the side length. Therefore, the volume of the wax in the first cube is \(0.5 * 6^3\).

Step 2: Compute the Volume of the Second Cube

Next, we find the volume of the second cube. The side length here is 4 inches. Since the cube is completely filled with wax, its volume is the same as that of the wax. Hence, the volume of the wax here is \(4^3\).

Step 3: Compare the Volumes

Now we need to compare the volumes of the wax in the two cubes. If the volume in the second cube is less than half the volume of the first cube, then it cannot hold the same amount of wax. If it is more, then it can.

Key Concepts

Cube Volume Calculation

Calculating the volume of a cube is a fundamental concept in geometry. A cube is a three-dimensional shape with six equal square faces. To find the volume of a cube, you need to know the length of one of its edges. You then multiply that length by itself three times because a cube has equal length, width, and height.
In this exercise, the first cube has each edge measuring 6 inches. The volume is calculated by cubing the edge length: \[ \text{Volume of the first cube} = 6 \times 6 \times 6 = 6^3 \text{ cubic inches.} \]
Since only half of this volume is used to hold the wax, we multiply the total volume by 0.5 to find the volume of the wax:
\[ \text{Volume of wax in the first cube} = 0.5 \times 6^3 = 108 \text{ cubic inches.} \]
Understanding how to calculate the volume of a cube is essential for solving similar geometry problems.

Volume Comparison

Comparing volumes helps determine if one container can hold the same amount of material as another. After calculating the volume of wax in the first cube, the next step is finding out the volume of the second cube.
The second cube's edge length is given as 4 inches. Therefore, its volume is:\[ \text{Volume of the second cube} = 4 \times 4 \times 4 = 4^3 = 64 \text{ cubic inches.} \]
Now, to determine if the second container can hold the same amount of wax as the first container, we compare the volumes:

• The volume of wax in the first container is 108 cubic inches.
• The total volume of the second container (fully filled) is just 64 cubic inches.

Clearly, 64 cubic inches is less than 108 cubic inches, so the second cube cannot hold the same amount of wax as the first container is holding in its half-filled state.

Geometry in Algebra

Geometry often involves algebra to solve real-world problems involving shapes and volumes. In this candle-making scenario, algebraic expressions are used to find the solutions quickly.
In both containers, you use the formula for the volume of a cube, which is an algebraic expression: \( s^3 \) where \( s \) is the side length. The problem involves setting up equations to find the volume and then comparing these values to solve the problem. This not only tests your understanding of geometric formulas but also your ability to manipulate algebraic expressions to make practical comparisons.
Besides straightforward volume calculations, such problems also prepare you to deal with more complex real-world situations. For instance, managing quantities, comparing sizes, and determining space efficiently are all skills honed by understanding how geometry and algebra intertwine. The use of algebraic formulas for geometric problems simplifies complex computations into more manageable algebraic tasks, providing tremendous help in visualization and problem-solving.