What Is The Length Of The Side Of A Cube That Has A Volume Of 0.001 Dam^3? The Options Are A) 1 Dm B) 1 M C) 1 Dam D) 1 Cm.

In the realm of geometry, understanding the relationship between volume and side length is crucial, especially when dealing with three-dimensional shapes like cubes. This article aims to delve into the process of determining the side length of a cube given its volume. Specifically, we will address the problem of finding the side length of a cube that has a volume of 0.001 cubic decameters (dam³). We will explore the fundamental concepts of volume calculation for cubes, unit conversions, and the step-by-step methodology to arrive at the correct answer. This exploration will not only provide a solution to the problem at hand but also enhance understanding of spatial reasoning and mathematical problem-solving in general. This discussion will be valuable for students, educators, and anyone interested in geometry and spatial calculations.

To effectively determine the side length of a cube, it is essential to first grasp the fundamental concept of volume. Volume, in three-dimensional geometry, refers to the amount of space a substance or object occupies. It is a scalar quantity usually expressed in cubic units, such as cubic meters (m³), cubic centimeters (cm³), or in this case, cubic decameters (dam³). A cube, being a regular hexahedron, is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. It is one of the five Platonic solids and the only regular hexahedron. All its edges are of the same length.

The volume of a cube is calculated by raising the length of one of its sides (or edges) to the power of three. Mathematically, this is represented as:

Volume (V) = s³

Where:

• V represents the volume of the cube.
• s represents the length of a side of the cube.

This formula is derived from the more general formula for the volume of a rectangular prism (length × width × height), where in a cube, the length, width, and height are all equal. Understanding this formula is the cornerstone of solving problems related to cube volumes and side lengths. The relationship is straightforward: if you know the side length, you can easily calculate the volume. Conversely, if you know the volume, you can determine the side length by taking the cube root of the volume. This inverse relationship is what we will use to solve the specific problem presented.

Furthermore, it's important to appreciate how the units of measurement play a crucial role in these calculations. If the side length is given in meters, the volume will be in cubic meters. Similarly, if the side length is in decameters, the volume will be in cubic decameters. This consistency in units is vital to avoid errors in calculations and interpretations. For instance, in our problem, the volume is given in cubic decameters, and we need to find the side length in a suitable unit, which may involve converting between different units of length.

Unit conversion is a critical skill in many scientific and mathematical contexts, and it is particularly relevant when dealing with volumes and lengths in geometry. In this specific problem, we are given the volume of a cube in cubic decameters (dam³), and to find the side length effectively, it's beneficial to convert this volume into a more commonly used unit, such as cubic decimeters (dm³) or cubic meters (m³). Understanding how to convert between these units is essential for accurate calculations and for comparing measurements in different scales.

To convert from cubic decameters (dam³) to cubic decimeters (dm³), we need to know the relationship between decameters and decimeters. One decameter (dam) is equal to ten meters (m), and one meter is equal to ten decimeters (dm). Therefore, one decameter is equal to 100 decimeters (1 dam = 10 m = 100 dm). However, we are dealing with cubic units, so we need to consider the cubic relationship:

1 dam³ = (1 dam)³ = (10 m)³ = 1000 m³

And since 1 m = 10 dm:

1 m³ = (10 dm)³ = 1000 dm³

Therefore:

1 dam³ = 1000 m³ = 1000 * 1000 dm³ = 1,000,000 dm³

Now, we can convert the given volume of 0.001 dam³ to dm³:

0. 001 dam³ = 0.001 * 1,000,000 dm³ = 1000 dm³

This conversion is a crucial step because it transforms the volume into a unit that is easier to visualize and work with when finding the side length. By converting to cubic decimeters, we can more readily relate the volume to the side length in decimeters, which is one of the options provided in the problem.

In addition to converting to dm³, it's also helpful to understand conversions to other units like cubic meters (m³) or cubic centimeters (cm³). Knowing these conversion factors allows for flexibility in problem-solving and ensures that the final answer is presented in the most appropriate unit. For example, converting to cubic meters would involve dividing the cubic decameter volume by 1000, while converting to cubic centimeters would involve multiplying by a larger factor, reflecting the smaller size of a cubic centimeter compared to a cubic decameter.

With the volume now converted to a more manageable unit, we can proceed to calculate the side length of the cube. This involves applying the inverse of the volume formula, which means finding the cube root of the volume. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, if V = s³, then s = ∛V.

In our case, we have determined that the volume of the cube is 1000 dm³. To find the side length (s), we need to calculate the cube root of 1000:

s = ∛1000 dm³

Finding the cube root of 1000 is a relatively straightforward calculation. We are looking for a number that, when multiplied by itself three times, equals 1000. We know that:

10 * 10 * 10 = 1000

Therefore, the cube root of 1000 is 10:

s = 10 dm

This result indicates that the side length of the cube is 10 decimeters. The calculation highlights the direct relationship between the volume of a cube and its side length, emphasizing how understanding the cube root operation is essential in solving such problems. The process of finding the cube root can sometimes involve more complex calculations, especially when dealing with non-perfect cubes. However, in this case, the number 1000 is a perfect cube, making the calculation simple and intuitive.

The result we've obtained, 10 dm, is a crucial piece of information that we will use to select the correct answer from the given options. It's important to remember the units throughout the calculation to ensure that the final answer is not only numerically correct but also dimensionally consistent. This step-by-step approach, from converting units to applying the cube root, demonstrates a methodical way to tackle geometry problems involving volume and side length.

Now that we have calculated the side length of the cube to be 10 decimeters (dm), the next crucial step is to compare our solution with the options provided in the problem. This involves ensuring that our answer matches one of the given choices and understanding the units used in each option. The options given are:

A) 1 dm B) 1 m C) 1 dam D) 1 cm

Our calculated side length is 10 dm. To determine which option matches our solution, we need to express all the options in the same unit or convert our solution to match one of the options. Let's examine each option:

• Option A) 1 dm: This option is in decimeters, the same unit as our calculated side length. However, 1 dm is not equal to our calculated 10 dm.
• Option B) 1 m: We know that 1 meter (m) is equal to 10 decimeters (dm). Therefore, 1 m is equal to our calculated 10 dm. This option appears to be a potential match.
• Option C) 1 dam: One decameter (dam) is equal to 10 meters or 100 decimeters. This is significantly larger than our calculated 10 dm.
• Option D) 1 cm: One centimeter (cm) is a much smaller unit than a decimeter (1 dm = 10 cm). Therefore, 1 cm is far smaller than our calculated 10 dm.

By comparing our solution with each option and converting units where necessary, it becomes clear that Option B) 1 m is the correct answer. Our calculated side length of 10 dm is equivalent to 1 meter, making it the only option that accurately represents the side length of the cube.

This process of matching the solution with the given options underscores the importance of unit consistency and the ability to convert between different units of measurement. It also highlights the significance of careful comparison to ensure that the final answer is not only mathematically correct but also correctly interpreted within the context of the problem.

In summary, the problem presented us with a cube having a volume of 0.001 dam³, and the task was to determine the length of one of its sides. Through a systematic approach, we successfully navigated the problem-solving process, demonstrating the key concepts of volume calculation, unit conversion, and cube root determination. We began by understanding the fundamental formula for the volume of a cube (V = s³) and the relationship between volume and side length. We then converted the given volume from cubic decameters to cubic decimeters, which simplified the subsequent calculations. This conversion was crucial, as it allowed us to work with a more manageable unit and directly apply the cube root to find the side length.

The calculation involved finding the cube root of 1000 dm³, which yielded a side length of 10 dm. Following this, we carefully compared our calculated side length with the provided options, ensuring that we accounted for the different units of measurement. This step highlighted the importance of unit consistency and the ability to convert between units accurately. Ultimately, we identified that 10 dm is equivalent to 1 meter, making Option B) 1 m the correct answer.

This exercise not only provided a solution to a specific geometric problem but also reinforced the broader principles of mathematical problem-solving. The step-by-step methodology employed, from understanding the problem statement to verifying the final answer, is a valuable approach that can be applied to a wide range of mathematical and scientific challenges. The ability to break down complex problems into smaller, more manageable steps, along with a solid understanding of fundamental concepts and mathematical operations, is key to success in these domains. Furthermore, the emphasis on unit conversion and dimensional analysis underscores the importance of precision and attention to detail in quantitative problem-solving. This holistic approach ensures not only the correctness of the answer but also a deeper understanding of the underlying principles and concepts involved.