Question

Which of the following rectangular metal boxes can hold the most dice if each die has a volume of one cubic inch and the lid of the box fits tightly? (A) 9 inches x 15 inches x 11 inches (B) 20 inches x 7.5 inches x 10 inches (C) 12.5 inches x 7.5 inches x 16.5 inches (D) 20 inches x 7.5 inches x 10.5 inches

Short answer

The rectangular metal box with the largest volume is Box D, having dimensions of 20 inches x 7.5 inches x 10.5 inches and a volume of 1575 cubic inches. Therefore, it can hold the most dice. The correct answer is option (D).

Step-by-step solution

Calculate the volume of Box A

To calculate the volume of Box A, we need to multiply its dimensions - length, width, and height: 9 inches × 15 inches × 11 inches. This gives us a volume of \(9 × 15 × 11 = 1485\) cubic inches.

Calculate the volume of Box B

To calculate the volume of Box B, we need to multiply its dimensions - length, width, and height: 20 inches × 7.5 inches × 10 inches. This gives us a volume of \(20 × 7.5 × 10 = 1500\) cubic inches.

Calculate the volume of Box C

To calculate the volume of Box C, we need to multiply its dimensions - length, width, and height: 12.5 inches × 7.5 inches × 16.5 inches. This gives us a volume of \(12.5 × 7.5 × 16.5 = 1546.875\) cubic inches.

Calculate the volume of Box D

To calculate the volume of Box D, we need to multiply its dimensions - length, width, and height: 20 inches × 7.5 inches × 10.5 inches. This gives us a volume of \(20 × 7.5 × 10.5 = 1575\) cubic inches.

Compare the volumes of boxes to find the largest

We have calculated the volumes of each box, and now it's time to compare the volumes: - Box A: 1485 cubic inches - Box B: 1500 cubic inches - Box C: 1546.875 cubic inches - Box D: 1575 cubic inches Box D has the largest volume at 1575 cubic inches, meaning it can hold the most dice. Hence, the correct answer is option (D).

Key concepts

GED math problems

GED, or General Educational Development tests, are designed to measure the skills and knowledge similar to a high school level curriculum. In the math section, students often encounter problems involving a variety of topics, including algebra, number operations, and geometry. The ability to approach these problems requires both a solid understanding of mathematical procedures and the skills to apply them in practical questions.

For example, when faced with a GED math problem involving volume, like the one with the rectangular metal boxes, students must recall the formula for the volume of a rectangular prism (V = lwh) where V is volume, length is , width is , and height is . It's not just about knowing the formula but also understanding what it represents: the amount of space within the box. This understanding will contribute to correctly solving the problem and choosing the appropriate dimensions when comparing the volumes of different boxes.

Geometry problems

Geometry problems in math often involve shapes, sizes, relative positions of figures, and the properties of space. A common type of geometry problem that students encounter in textbooks is calculating the volume of three-dimensional figures like prisms, cylinders, spheres, and cones.

Understanding the formulas for the volume of various shapes is crucial. A rectangular box, which is a type of prism, has a simple volume calculation that involves multiplying its length, width, and height. The exercise provided is a practical application of this geometric formula. Students not only need to apply the formula but must also understand spatial reasoning, which allows them to visualize the problem and ensure that they use the correct dimensions while performing calculations.

Mathematical reasoning

Mathematical reasoning is the logical thinking process used to solve math problems. It involves recognizing patterns, constructing logical arguments, and making deductions. In the context of solving volume problems, mathematical reasoning is applied by understanding the relationship between the different dimensions and how they affect the overall volume.

When working through the provided volume calculation exercise, students practice mathematical reasoning by setting up an equation for each box and solving for volume. The reasoning extends to comparing the results and determining which box holds the most dice. This step is critical because identifying the box with the highest volume isn't just about calculating numbers but about understanding what these numbers represent in the real world.

Volume comparison

Volume comparison problems require students to analyze the capacity of different objects and determine which one can contain more or less of a substance or items, like the dice in the provided problem. This type of problem enhances students' critical thinking and ability to apply their knowledge to real-life situations.

By computing the volumes of several boxes and comparing them, students engage in the process of volume comparison. Key to these comparisons is not only performing arithmetic accurately but also ensuring measurements are used correctly. Students must also pay attention to units of measure and convert them if necessary to make valid comparisons. The comparison is often the final step in volume problems and is where students demonstrate full comprehension of the volume concept by interpreting the calculations properly to arrive at a solution.