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

Step 1: Understand the Given Problem

We need to find the difference in volume between a cube with side length \(x+2\) and a cube with side length \(x\). We will use the Binomial Theorem to expand \((x + 2)^3\) and find this difference.

Step 2: Recall the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}\). In our problem, \(a = x\), \(b = 2\), and \(n = 3\).

Step 3: Apply the Binomial Theorem

Using the Binomial Theorem, we express \((x + 2)^3\) as follows:\[(x + 2)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} 2^k\] =\[\binom{3}{0} x^3 2^0 + \binom{3}{1} x^2 2^1 + \binom{3}{2} x^1 2^2 + \binom{3}{3} x^0 2^3\].

Step 4: Simplify the Expansion

Calculate each term in the expansion:- \(\binom{3}{0} x^3 2^0 = x^3\)- \(\binom{3}{1} x^2 2^1 = 3x^2 \times 2 = 6x^2\)- \(\binom{3}{2} x 2^2 = 3x \times 4 = 12x\)- \(\binom{3}{3} 2^3 = 8\)Thus, \((x + 2)^3 = x^3 + 6x^2 + 12x + 8\).

Step 5: Calculate the Difference in Volumes

The volume difference between the larger cube \((x + 2)^3\) and the smaller cube \(x^3\) is:\[(x+2)^3 - x^3 = (x^3 + 6x^2 + 12x + 8) - x^3\]After canceling \(x^3\), we get:\[6x^2 + 12x + 8\].

Step 6: Verify the Result

The difference in volume calculated is exactly \(6x^2 + 12x + 8\). This matches the problem's requirement, confirming our expansion and calculation are correct.

Key Concepts

Volume of a Cube

A cube is a three-dimensional shape with six equal square faces. The volume of a cube is the capacity it can hold within its boundaries. To calculate the volume, you multiply the side length by itself twice. This is because a cube has equal length, width, and height.

The formula for the volume of a cube with side length \(x\) is given by \(V(x) = x^3\). This formula is derived from the idea of multiplying the base area \(x \times x = x^2\) by the height, which is also \(x\).

So, when we have a cube with side \(x+2\), its volume is calculated similarly: \((x+2)^3\). This involves multiplying \((x+2)\) by itself twice more, leading to \((x+2)(x+2)(x+2)\). Understanding these basic principles is crucial before expanding the expression using the Binomial Theorem.

Polynomial Expansion

The polynomial expansion is a key concept that helps in simplifying expressions such as \((x+2)^3\). An essential tool for this is the Binomial Theorem. It allows us to expand expressions raised to a power in a systematic way. The theorem is stated as: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\),where \(n\) is a positive integer and \(\binom{n}{k}\) is the binomial coefficient.

In the context of the exercise, we apply it to \((x+2)^3\):

• Here, \(a = x\), \(b = 2\), and \(n = 3\).
• We calculate each term: \(\binom{3}{0}x^3 \cdot 2^0\), \(\binom{3}{1}x^2 \cdot 2^1\), \(\binom{3}{2}x^1 \cdot 2^2\), \(\binom{3}{3}x^0 \cdot 2^3\).
• When simplified, it becomes \(x^3 + 6x^2 + 12x + 8\).

Polynomial expansion is thus an effective way to break down complex expressions, making calculations and comparisons easier, such as when determining the change in volume of cubes.

Difference in Volume

When considering the difference in the volumes of two cubes, it refers to how much more space the larger cube occupies compared to the smaller one. This can naturally occur when you increase the side length of the cube. Applying this to our problem:

• The larger cube has a side of \(x+2\), giving it a volume of \((x+2)^3\).
• The smaller cube has a side of \(x\), yielding \(x^3\) as its volume.

To find the difference:

• Subtract the smaller cube's volume from the larger cube's volume: \((x+2)^3 - x^3\).
• Using our polynomial expansion, this becomes: \[(x^3 + 6x^2 + 12x + 8) - x^3\].
• Simplifying results in \(6x^2 + 12x + 8\).

This outcome reflects the precise increment in volume as the cube's side length increases by 2 units. Understanding this concept is valuable in real-world calculations where changes in dimensions affect space and capacity.