Problem 55 - Precalculus Mathematics for Calculus

Question

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 in volume is \(6x^2 + 12x + 8\).

Step 1: Understand the Problem

We are given the volumes of two cubes, one with side length \(x\) inches and another with side length \(x+2\) inches. Our goal is to find the difference between these volumes: \(V(x+2) - V(x)\), and show that it simplifies to \(6x^2 + 12x + 8\).

Step 2: Use the Binomial Theorem

The Binomial Theorem states \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, we want to expand \((x+2)^3\). According to the theorem:\[(x+2)^3 = x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2 + 2^3 = x^3 + 6x^2 + 12x + 8\].

Step 3: Find the Volume Difference

Now, calculate \(V(x+2) - V(x)\). We have:\[V(x) = x^3\] and \[V(x+2) = x^3 + 6x^2 + 12x + 8\].Subtract these:\[V(x+2) - V(x) = (x^3 + 6x^2 + 12x + 8) - x^3 = 6x^2 + 12x + 8\].

Step 4: Verify the Result

The difference \(6x^2 + 12x + 8\) matches the expression given in the problem statement. Therefore, we have demonstrated using the Binomial Theorem that the difference in volume between the two cubes simplifies to this expression.

Key Concepts

Volume of a Cube

A cube is a three-dimensional shape with equal sides. The formula for the volume of a cube with side length \(x\) is given by \(V(x) = x^3\). This means you multiply the side length by itself three times.
Calculating the volume helps understand how much space the cube occupies.

• Small changes in the side length can significantly impact the volume due to cubing the side length.

Understanding the basic formula for a cube's volume sets the foundation for more complex problems, such as comparing the volumes of two cubes with different side lengths.

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra where expressions are expanded to reveal their terms. In our case, expanding \((x+2)^3\) involves writing out each term explicitly by cubing the binomial expression.
This process is crucial for simplifying expressions into a form where you can easily work with individual terms.

• It helps identify each component of the polynomial by performing operations like multiplication.

Having expanded the polynomial into terms such as \(x^3, 6x^2, 12x,\) and \(8\), we can more easily analyze and simplify the expression.
Polynomial expansion is an essential step before using methods like factoring or simplification.

Binomial Expansion

The binomial expansion is derived from the Binomial Theorem, which helps expand expressions raised to a power. It states that \((a + b)^n\) can be expanded using combinations and powers of \(a\) and \(b\).
For example, expanding \((x+2)^3\) gives \(x^3 + 3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2 + 2^3\).

• Every term in the expansion has a coefficient; the coefficients are combinations \(\binom{n}{k}\) from Pascal's Triangle.
• Each term's powers add up to \(n\), the original exponent.

This approach simplifies complex power calculations into manageable steps, ensuring you accurately expand any binomial expression.

Differences in Volume

When comparing volumes of two cubes with different side lengths, we use polynomial operations to find the difference. Given the volumes \(V(x) = x^3\) and \(V(x+2) = (x+2)^3\), the difference is \(V(x+2) - V(x)\).
After expanding and simplifying \((x+2)^3\), we simply subtract \(x^3\):

• This gives a formula for the extra volume: \(6x^2 + 12x + 8\).
• It shows how an increment in side length affects the volume considerably.

Understanding these differences helps illustrate the profound impact of even small increases in dimensions on volume in three-dimensional shapes.