Cube Volume Calculation Expanding (x - 2y)³

#Problem The side length, s, of a cube is x - 2y. If V = s³, what is the volume of the cube?

A. x³ - 6x²y + 12xy² - 8y³ B. x³ + 6x²y + 12xy² + 8y³ C. 3x³ - 6x²y + 12xy² - 24y³ D. x³ - 8y³

Solution

To find the volume of the cube, we need to calculate V = s³, where s = x - 2y. This means we need to expand (x - 2y)³. Let's delve into the step-by-step solution, ensuring a clear understanding of the algebraic expansion involved.

Step 1: Understanding the Problem and the Formula

In this mathematical puzzle, we're presented with a cube whose side length, denoted as s, is expressed algebraically as x - 2y. Our mission is to determine the cube's volume, V, using the fundamental formula V = s³. This task requires us to expand the binomial expression (x - 2y)³. Before we jump into the expansion, let's recap the binomial theorem or the algebraic identity that will serve as our cornerstone for solving this problem. The identity we're interested in is the cube of a binomial difference, which states:

(a - b)³ = a³ - 3a²b + 3ab² - b³

This formula provides a structured way to expand expressions of the form (a - b)³, which is precisely what we need to tackle our problem. Understanding and correctly applying this identity is crucial for navigating through the algebraic manipulations ahead. Now that we have our formula ready, let's proceed to the next step, where we'll carefully substitute the given values into the formula and begin the expansion process.

Step 2: Applying the Binomial Cube Formula

Now, let's apply the binomial cube formula to our specific problem. We have s = x - 2y, so we need to find (x - 2y)³. Using the formula (a - b)³ = a³ - 3a²b + 3ab² - b³, we substitute a with x and b with 2y. This substitution is a pivotal step in bridging the gap between the general formula and our specific algebraic expression. The careful replacement of variables ensures that we're setting up the equation correctly for expansion. Now, let's perform this substitution and lay out the expanded form before any simplification:

(x - 2y)³ = x³ - 3(x²)(2y) + 3(x)(2y)² - (2y)³

This expanded form is a direct application of the binomial cube formula, showcasing each term's contribution. The next step involves simplifying each term to reveal the final expanded polynomial. Let's move on to simplifying this expression, where we'll perform the necessary arithmetic operations to condense the terms and bring us closer to the solution.

Step 3: Simplify the Expression

With the expression expanded, our next task is to simplify each term. This involves performing the multiplications and exponentiations indicated in the expression. Let's break down the simplification step by step to ensure clarity.

First, we address the term -3(x²)(2y). Multiplying the constants gives us -6, so this term simplifies to -6x²y. Next, we simplify the term 3(x)(2y)². We first square 2y to get 4y², then multiply by 3x to obtain 12xy². Finally, we simplify the term (2y)³. Cubing 2y means multiplying it by itself three times, which yields 8y³. Now, let's put these simplified terms back into the expression:

x³ - 6x²y + 12xy² - 8y³

This simplified expression represents the volume V of the cube in terms of x and y. It's a polynomial expression, where each term contributes to the overall volume based on the values of x and y. Comparing this result with the given options, we can identify the correct answer. The simplification process is a critical step in solving algebraic problems, as it transforms the expanded form into a more manageable and recognizable expression.

Step 4: Identify the Correct Option

After simplifying the expression, we arrive at the volume of the cube as:

V = x³ - 6x²y + 12xy² - 8y³

Now, we compare this result with the multiple-choice options provided in the problem statement. By careful observation, we can see that our simplified expression exactly matches option A:

A. x³ - 6x²y + 12xy² - 8y³

Therefore, option A is the correct answer. This step is crucial in the problem-solving process, as it confirms that our algebraic manipulations and simplifications have led us to the accurate solution. Matching the derived expression with the given options provides a sense of validation and ensures that we're on the right track.

Conclusion

Therefore, the volume of the cube is x³ - 6x²y + 12xy² - 8y³, which corresponds to option A. This problem elegantly combines the geometric concept of a cube's volume with algebraic expansion, specifically the binomial cube formula. The step-by-step solution illustrates how applying the correct algebraic identity and carefully simplifying the resulting expression leads to the answer. Understanding such connections between geometry and algebra is fundamental in mathematical problem-solving. The solution not only provides the correct answer but also reinforces the importance of algebraic manipulation skills in various mathematical contexts.

Final Answer: The final answer is (A)