Problem 55 - Algebra and Trigonometry

Question

Difference in Volumes of Cubes The volume of a cube of side \(x\) inches is given by \(V(x)=x^{3},\) so the volume of a cube of side \(X+2\) inches is given by \(V(x+2)=(x+2)^{3}\) . Use the Binomial Theorem to show that the difference in volume between the larger and smaller cubes is \(6 x^{2}+12 x+8\) cubic inches

Step-by-Step Solution

Answer

The difference is \(6x^2 + 12x + 8\) cubic inches.

Step 1: Understanding the Problem

The problem asks us to find the difference in volume between a larger cube with side length \(x+2\) and a smaller cube with side length \(x\). We are given the formulas for their volumes: \(V(x) = x^3\) and \(V(x+2) = (x+2)^3\). The goal is to express the difference \(V(x+2) - V(x)\) as a polynomial using the Binomial Theorem.

Step 2: Applying the Binomial Theorem

The Binomial Theorem states that \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Let's substitute \(a = x\) and \(b = 2\) into this formula to expand \((x+2)^3\).

Step 3: Expanding \((x+2)^3\)

Using the Binomial Theorem, we expand:\[(x+2)^3 = x^3 + 3(x^2)(2) + 3(x)(2^2) + 2^3\]Simplifying, this becomes:\[x^3 + 6x^2 + 12x + 8\]

Step 4: Calculating the Difference in Volumes

The difference in volumes is given by subtracting the volume of the smaller cube from the volume of the larger cube:\[(x+2)^3 - x^3 = (x^3 + 6x^2 + 12x + 8) - x^3\]This simplifies to:\[6x^2 + 12x + 8\]

Step 5: Conclusion

The difference in volumes between the cube with side \(x+2\) and the cube with side \(x\) is \(6x^2 + 12x + 8\), confirming the expression provided in the exercise.

Key Concepts

Volume of a Cube

To understand the volume of a cube, envision a six-sided solid where all sides are equal in length. If we label the side length as \( x \), then the volume is calculated by the formula \( V(x) = x^3 \). This simply means the side length raised to the power of three. It's like multiplying the length, width, and height, which are all the same in a cube. Knowing the volume formula is crucial because it is the starting point for solving problems related to cubes.
Consider a cube with a side length of \( x+2 \). Now, the formula becomes \( V(x+2) = (x+2)^3 \). This represents another cube that is slightly larger, with each side extended by 2 units compared to the first cube.
Understanding how the volume of a cube changes with a small increase in side length helps when manipulating expressions involving cubes in algebra. Remember, volume is always expressed in cubic units because it measures three-dimensional space.

Polynomial Expansion

Polynomial expansion, especially in the context of binomials, is a powerful tool in algebra. The Binomial Theorem simplifies expanding powers of sums, such as \((a+b)^n\). Here, we specifically used it for \((x+2)^3\), where \(a = x\) and \(b = 2\) with \(n = 3\).
The theorem provides a formula:

• The first term is \(a^3\), or \(x^3\).
• Next, \(3a^2b\) becomes \(3x^2 \times 2\).
• The third term, \(3ab^2\), simplifies to \(3x \times 2^2\).
• Finally, add \(b^3\), resulting in \(2^3 = 8\).

When expanded, the expression is \((x+2)^3 = x^3 + 6x^2 + 12x + 8\). Each term arises naturally from substituting the variables into the binomial formula. Polynomial expansion allows understanding complex algebraic structures and how they grow with different variables.

Difference of Cubes

The difference of cubes is found by subtracting one cube's volume from another's. In algebra, this difference can be expressed in a structured way using the formula. For cubes with side lengths \(x\) and \(x+2\), the volumes are \(x^3\) and \((x+2)^3\) respectively.
To find the difference of these cubes' volumes, use:

• First, expand \((x+2)^3\) using the Binomial Theorem as shown: \(x^3 + 6x^2 + 12x + 8\).
• Then subtract the smaller cube's volume: \((x+2)^3 - x^3 = x^3 + 6x^2 + 12x + 8 - x^3\).
• This simplifies neatly to: \(6x^2 + 12x + 8\).

This formula reveals the difference due to the increased size of each dimension of the larger cube compared to the smaller one. The result shows how small increments in cube side lengths can result in significant growth in the volume.