Question
A cube is shrunk so that its side lengths of 2 are reduced by 3 units. Using the binomial theorem, what is the correct expression for the new volume of the cube?
Answers
Answered by
GPT-4o mini
To find the new volume of the cube after its side lengths are reduced, we can start with the original side length of the cube, which is 2 units.
The original volume of the cube is given by:
\[
V = s^3
\]
where \( s \) is the side length of the cube. For \( s = 2 \),
\[
V_{\text{original}} = 2^3 = 8.
\]
Now, we want to reduce the side length by 3 units. The new side length \( s' \) will be:
\[
s' = 2 - 3 = -1.
\]
Since the side length cannot be negative, this initially seems problematic. Instead, we should reinterpret the meaning of the problem. The reduction of 3 units cannot be applied directly here since the side length cannot be less than zero. We can infer that this might imply a change in the perspective of cube dimensions for any valid size.
Instead, let’s consider the term "reduce by 3 units" more generally in relation to calculating the volume using the binomial theorem. For any side length expressed as \( s = 2 \), and \( x = -3 \):
We can express the side length after applying a reduction of 3 units generally as:
\[
s' = 2 +(-3) = 2 + x.
\]
In this case \( x = -3 \). We can now use the binomial theorem to express \( (s + x)^3 \).
Using the binomial theorem:
\[
(s + x)^3 = s^3 + 3s^2x + 3sx^2 + x^3.
\]
Substituting \( s = 2 \) and \( x = -3 \):
\[
(2 - 3)^3 = 2^3 + 3(2^2)(-3) + 3(2)(-3)^2 + (-3)^3.
\]
Now calculating each term:
- The first term: \( 2^3 = 8 \)
- The second term: \( 3(2^2)(-3) = 3(4)(-3) = -36 \)
- The third term: \( 3(2)(-3)^2 = 3(2)(9) = 54 \)
- The fourth term: \( (-3)^3 = -27 \)
Now, summing these contributes to the new volume:
\[
V' = 8 - 36 + 54 - 27.
\]
Calculating this step-by-step:
\[
V' = 8 - 36 = -28,
\]
\[
-28 + 54 = 26,
\]
\[
26 - 27 = -1.
\]
Thus, **the new volume** of the cube, albeit mathematically computed, leads us to conclude a non-physical negative volume. However, that considered mathematically you will see:
\[
\text{The correct expression for the new volume using the binomial theorem is} \ (2 - 3)^3 = -1 \text{ (implying the cube is physically invalid with negative side)}.
\]
This illustrates \( (s - 3 )^3 \) evidently leading to reinterpretation discussing dimensions and requirement in scenarios beyond zero. Hence if required rethink practical dimensional setups or side limits.
The original volume of the cube is given by:
\[
V = s^3
\]
where \( s \) is the side length of the cube. For \( s = 2 \),
\[
V_{\text{original}} = 2^3 = 8.
\]
Now, we want to reduce the side length by 3 units. The new side length \( s' \) will be:
\[
s' = 2 - 3 = -1.
\]
Since the side length cannot be negative, this initially seems problematic. Instead, we should reinterpret the meaning of the problem. The reduction of 3 units cannot be applied directly here since the side length cannot be less than zero. We can infer that this might imply a change in the perspective of cube dimensions for any valid size.
Instead, let’s consider the term "reduce by 3 units" more generally in relation to calculating the volume using the binomial theorem. For any side length expressed as \( s = 2 \), and \( x = -3 \):
We can express the side length after applying a reduction of 3 units generally as:
\[
s' = 2 +(-3) = 2 + x.
\]
In this case \( x = -3 \). We can now use the binomial theorem to express \( (s + x)^3 \).
Using the binomial theorem:
\[
(s + x)^3 = s^3 + 3s^2x + 3sx^2 + x^3.
\]
Substituting \( s = 2 \) and \( x = -3 \):
\[
(2 - 3)^3 = 2^3 + 3(2^2)(-3) + 3(2)(-3)^2 + (-3)^3.
\]
Now calculating each term:
- The first term: \( 2^3 = 8 \)
- The second term: \( 3(2^2)(-3) = 3(4)(-3) = -36 \)
- The third term: \( 3(2)(-3)^2 = 3(2)(9) = 54 \)
- The fourth term: \( (-3)^3 = -27 \)
Now, summing these contributes to the new volume:
\[
V' = 8 - 36 + 54 - 27.
\]
Calculating this step-by-step:
\[
V' = 8 - 36 = -28,
\]
\[
-28 + 54 = 26,
\]
\[
26 - 27 = -1.
\]
Thus, **the new volume** of the cube, albeit mathematically computed, leads us to conclude a non-physical negative volume. However, that considered mathematically you will see:
\[
\text{The correct expression for the new volume using the binomial theorem is} \ (2 - 3)^3 = -1 \text{ (implying the cube is physically invalid with negative side)}.
\]
This illustrates \( (s - 3 )^3 \) evidently leading to reinterpretation discussing dimensions and requirement in scenarios beyond zero. Hence if required rethink practical dimensional setups or side limits.