A cube is shrunk so that its side lengths of 2 are reduced by 3 units. Using the binomial theorem, which of the following is the correct expression for the new volume of the cube?

Question

A cube is shrunk so that its side lengths of 2 are reduced by 3 units. Using the binomial theorem, which of the following is the correct expression for the new volume of the cube?
a. 8x^3+36x^2+54x-27
b. 8x^3-36x^2+54x+27
c. 8x^3+36x^2+54x+27
d. 8x^3-36x^2+54x-27

GPT-4o mini · Answer

To solve the problem, we start by noting that the original side length of the cube is 2 units. We want to shrink this side length by 3 units. Since the original side length is 2, reducing it by 3 units would actually lead to a new side length of:

$2 - 3 = - 1$

This indicates that the cube effectively has a negative side length, which physically doesn’t exist. However, mathematically, we can still explore the volume of a cube with a negative side length.

The volume $V$ of a cube with side length $s$ is given by:

$V = s^{3}$

Substituting our new side length $s = - 1$ , we find:

$V = (- 1)^{3} = - 1$

However, it appears that the intention of the question is to express the new volume as a binomial expansion.

Let’s express the original volume $V = s^{3}$ using the binomial theorem where:

$s = 2 - x$
$x$ represents the amount by which the side length is decreased.

Here, we want to perform the expansion around $x = 3$ :

$(2 - x)^{3}$

We can apply the binomial theorem:

$(a + b)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) a^{n - k} b^{k}$

For our case, set $a = 2$ and $b = - x$ with $n = 3$ :

$(2 - x)^{3} = \sum_{k = 0}^{3} (\binom{3}{k}) 2^{3 - k} (- x)^{k}$

Calculating each term individually:

For $k = 0$ : $(\binom{3}{0}) 2^{3} (- x)^{0} = 1 \cdot 8 \cdot 1 = 8$
For $k = 1$ : $(\binom{3}{1}) 2^{2} (- x)^{1} = 3 \cdot 4 \cdot (- x) = - 12 x$
For $k = 2$ : $(\binom{3}{2}) 2^{1} (- x)^{2} = 3 \cdot 2 \cdot x^{2} = 6 x^{2}$
For $k = 3$ : $(\binom{3}{3}) 2^{0} (- x)^{3} = 1 \cdot 1 \cdot (- x^{3}) = - x^{3}$

Adding these terms together,

$(2 - x)^{3} = 8 - 12 x + 6 x^{2} - x^{3}$

Now, substituting $x = 3$ :

$(2 - 3)^{3} = (- 1)^{3} = - 1$

However, let's examine our initial choices to find the one that aligns with the form derived here. The volume of a cube is cubic in terms of the shrinking forward. To interpret potential θ terms, we shall also inspect closely.

None of the provided expression seems to match (with each expression having differing coefficients). For validity, let's summarize the preferred cubic representation $(2 - x)^{3}$ leading us to:

$8 - 12 x + 6 x^{2} - x^{3}$

Guaranteed $x = 3$ , however finding designated matches would require extracting both continuously. On analyzing:

None directly show exact match to parameters, but terms hint and definitions suggest $d . 8 x^{3} - 36 x^{2} + 54 x - 27$ unless reassessment on distances pursued further towards paces.

Conclusively, running these through base parameters should lead where direct computations remained vigilant yet, match beyond engagements appears inevitable further questions shall fill gaps.

Verifying each aspect returns none listed read exacts, ensuring demands continue as per $V = (2 - x)^{3}$ returns otherwise consistent with continued exploratives.