Calculating Cube Side Length Differences Using Surface Area Formula

This article delves into a mathematical problem concerning the relationship between a cube's surface area and the length of its sides. Specifically, we will explore how to calculate the difference in side lengths between two cubes with differing surface areas. The formula provided, s = √ (SA/6), is crucial for determining the side length (s) of a cube given its surface area (SA). Our goal is to understand and apply this formula to solve a practical problem: finding out how much longer the side of a cube with a surface area of 84 square meters is compared to a cube with a surface area of 48 square meters. This exploration will not only enhance our understanding of geometric relationships but also highlight the application of mathematical formulas in real-world scenarios.

The cornerstone of our calculations is the formula s = √ (SA/6). Let's break this down to ensure a comprehensive understanding. In this equation, s represents the length of one side of the cube, a fundamental dimension that dictates the cube's overall size. The variable SA stands for the total surface area of the cube, measured in square units. The number 6 is significant because a cube has six congruent square faces. Therefore, when we divide the total surface area by 6 (SA/6), we are essentially finding the area of one of these square faces. The square root operation (√) then comes into play. By taking the square root of the area of one face, we arrive at the length of one side of that square, which, in turn, is the side length (s) of the entire cube.

This formula elegantly encapsulates the relationship between a cube's surface area and its side length. It's a direct consequence of the geometry of a cube and the way surface area is calculated. To further clarify, consider a cube with a side length of s. Each face has an area of s², and since there are six faces, the total surface area (SA) is 6s². Rearranging this equation to solve for s gives us s = √ (SA/6). Understanding the derivation of this formula enhances our confidence in its accuracy and applicability. It's not just a magical equation; it's a logical representation of geometric principles. With this foundational knowledge, we can confidently tackle the problem at hand, calculating the side lengths of cubes with given surface areas and determining their difference. This formula is a powerful tool for anyone working with three-dimensional geometry, providing a simple yet effective way to relate a cube's surface area to its linear dimensions.

Step-by-Step Solution

To determine how much longer the side of a cube with a surface area of 84 square meters is than a cube with a surface area of 48 square meters, we will follow a step-by-step approach:

Step 1: Calculate the side length of the cube with a surface area of 84 square meters.

Using the formula s = √ (SA/6), we substitute SA with 84:

s₁ = √ (84/6)

s₁ = √ 14

s₁ ≈ 3.74 meters

Therefore, the side length of the cube with a surface area of 84 square meters is approximately 3.74 meters.

Step 2: Calculate the side length of the cube with a surface area of 48 square meters.

Again, using the formula s = √ (SA/6), this time we substitute SA with 48:

s₂ = √ (48/6)

s₂ = √ 8

s₂ ≈ 2.83 meters

Thus, the side length of the cube with a surface area of 48 square meters is approximately 2.83 meters.

Step 3: Find the difference in side lengths.

To find out how much longer the side of the larger cube is, we subtract the side length of the smaller cube from the side length of the larger cube:

Difference = s₁ - s₂

Difference ≈ 3.74 - 2.83

Difference ≈ 0.91 meters

Therefore, the side of the cube with a surface area of 84 square meters is approximately 0.91 meters longer than the side of the cube with a surface area of 48 square meters.

Detailed Calculations Explained

In the previous section, we outlined the step-by-step solution to the problem. Here, we will delve deeper into the calculations to provide a more comprehensive understanding of the process. This detailed explanation will not only reinforce the steps but also offer insights into the mathematical reasoning behind each calculation.

Step 1 Detailed Explanation: Calculating the Side Length for SA = 84 square meters

We begin with the formula s = √ (SA/6), which, as we've established, relates the side length of a cube to its surface area. Our first task is to determine the side length (s₁) of the cube with a surface area (SA) of 84 square meters. Substituting SA with 84 in the formula, we get:

s₁ = √ (84/6)

The first operation we perform is the division inside the square root. Dividing 84 by 6 yields 14:

s₁ = √ 14

Now, we need to find the square root of 14. Since 14 is not a perfect square, its square root is an irrational number, meaning it cannot be expressed as a simple fraction. We can approximate its value using a calculator or through estimation. The square root of 14 is approximately 3.74165738677...

For practical purposes, we round this to two decimal places, giving us:

s₁ ≈ 3.74 meters

This result tells us that each side of the cube with a surface area of 84 square meters is approximately 3.74 meters long. This calculation is crucial as it forms the basis for comparing the side lengths of the two cubes. The precision we choose to maintain (in this case, two decimal places) depends on the level of accuracy required for the specific application. Understanding the steps involved in this calculation, from substituting the surface area into the formula to approximating the square root, is fundamental to solving similar problems.

Step 2 Detailed Explanation: Calculating the Side Length for SA = 48 square meters

Having calculated the side length of the cube with a surface area of 84 square meters, we now turn our attention to the cube with a surface area of 48 square meters. We follow the same procedure, applying the formula s = √ (SA/6). This time, we substitute SA with 48, representing the surface area of the second cube:

s₂ = √ (48/6)

As before, the first operation is the division within the square root. Dividing 48 by 6 results in 8:

s₂ = √ 8

Now, we need to calculate the square root of 8. Similar to 14, 8 is not a perfect square, so its square root is also an irrational number. The square root of 8 can be approximated as 2.82842712475...

Rounding this value to two decimal places for consistency with our previous calculation, we get:

s₂ ≈ 2.83 meters

This tells us that the side length of the cube with a surface area of 48 square meters is approximately 2.83 meters. This value is essential for our final step, where we will compare it to the side length of the other cube. Understanding how we arrived at this result, by applying the formula and approximating the square root, is key to grasping the mathematical concepts involved. The process highlights the importance of accurate calculation and the use of approximation when dealing with irrational numbers.

Step 3 Detailed Explanation: Finding the Difference in Side Lengths

With the side lengths of both cubes calculated, the final step is to determine the difference between them. This will tell us how much longer the side of the cube with a surface area of 84 square meters is compared to the cube with a surface area of 48 square meters. We have already found that:

• The side length of the cube with a surface area of 84 square meters (s₁) is approximately 3.74 meters.
• The side length of the cube with a surface area of 48 square meters (s₂) is approximately 2.83 meters.

To find the difference, we subtract the smaller side length from the larger side length:

Difference = s₁ - s₂

Substituting the values we calculated:

Difference ≈ 3.74 - 2.83

Performing the subtraction:

Difference ≈ 0.91 meters

This result indicates that the side of the cube with a surface area of 84 square meters is approximately 0.91 meters longer than the side of the cube with a surface area of 48 square meters. This final calculation answers the original question and demonstrates the practical application of the formula s = √ (SA/6). It also highlights the importance of understanding the units of measurement (meters in this case) and interpreting the result in the context of the problem. The difference in side lengths gives us a tangible measure of the size disparity between the two cubes.

In conclusion, by applying the formula s = √ (SA/6), we successfully determined that the side of a cube with a surface area of 84 square meters is approximately 0.91 meters longer than the side of a cube with a surface area of 48 square meters. This exercise demonstrates the practical application of mathematical formulas in solving real-world geometric problems. Understanding the relationship between a cube's surface area and its side length allows us to make comparisons and calculations that are relevant in various fields, from engineering to design. The step-by-step approach we followed, from understanding the formula to performing detailed calculations, provides a clear methodology for tackling similar problems. This exploration not only reinforces our mathematical skills but also enhances our ability to analyze and interpret geometric relationships.