Question
. The area of a rectangle is 8x ^ 2 - 32 The length of the rectangle is x - 2 Which of the following expressions could represent the width of the rectangle?
A. 4(x - 2)
B. 4(x - 2)
C 2(x - 4)
D. 2(x - 4)
A. 4(x - 2)
B. 4(x - 2)
C 2(x - 4)
D. 2(x - 4)
Answers
GPT-4o mini
To find the width of the rectangle, we can use the formula for the area of a rectangle:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
In this case, the area of the rectangle is given as \(8x^2 - 32\), and the length is given as \(x - 2\). We can denote the width as \(W\). Thus, we have:
\[
8x^2 - 32 = (x - 2) \times W
\]
To find \(W\), we can isolate it:
\[
W = \frac{8x^2 - 32}{x - 2}
\]
Next, we can factor the area \(8x^2 - 32\):
\[
8x^2 - 32 = 8(x^2 - 4) = 8(x - 2)(x + 2)
\]
Now we can substitute this back into the equation for \(W\):
\[
W = \frac{8(x - 2)(x + 2)}{x - 2}
\]
As long as \(x \neq 2\), we can cancel \(x - 2\):
\[
W = 8(x + 2)
\]
Now that we have \(W\), let's see which of the provided options could represent the width.
Looking at the answer options:
A. \(4(x - 2)\)
B. \(4(x - 2)\)
C. \(2(x - 4)\)
D. \(2(x - 4)\)
None of the options seem to match \(8(x + 2)\). However, if we have to assess transformations or equivalent forms between the answers given, we could simplify \(8(x + 2)\) or look for equivalent factors.
Since most of the options provided involve some forms of factored expressions, none directly match, indicating there may be an issue in terms of equality or given parameters.
Still, \(4(x - 2) = 4x - 8\) and \(2(x - 4) = 2x - 8\) will not yield \(8(x + 2) = 8x + 16\), and none of these are equivalent to our width calculation.
As it stands, none of the options A, B, C, or D directly represent \(W\), but \(8(x + 2)\) is equivalent to \(8x + 16\).
Upon further checking options, since any external factors leading to simplifications or transformations might present \(W\) in an unforeseen manner, direct existing values of wfactors do not equate accurately. Please check that provided values are accurate or for any preconditions.
\[
\text{Area} = \text{Length} \times \text{Width}
\]
In this case, the area of the rectangle is given as \(8x^2 - 32\), and the length is given as \(x - 2\). We can denote the width as \(W\). Thus, we have:
\[
8x^2 - 32 = (x - 2) \times W
\]
To find \(W\), we can isolate it:
\[
W = \frac{8x^2 - 32}{x - 2}
\]
Next, we can factor the area \(8x^2 - 32\):
\[
8x^2 - 32 = 8(x^2 - 4) = 8(x - 2)(x + 2)
\]
Now we can substitute this back into the equation for \(W\):
\[
W = \frac{8(x - 2)(x + 2)}{x - 2}
\]
As long as \(x \neq 2\), we can cancel \(x - 2\):
\[
W = 8(x + 2)
\]
Now that we have \(W\), let's see which of the provided options could represent the width.
Looking at the answer options:
A. \(4(x - 2)\)
B. \(4(x - 2)\)
C. \(2(x - 4)\)
D. \(2(x - 4)\)
None of the options seem to match \(8(x + 2)\). However, if we have to assess transformations or equivalent forms between the answers given, we could simplify \(8(x + 2)\) or look for equivalent factors.
Since most of the options provided involve some forms of factored expressions, none directly match, indicating there may be an issue in terms of equality or given parameters.
Still, \(4(x - 2) = 4x - 8\) and \(2(x - 4) = 2x - 8\) will not yield \(8(x + 2) = 8x + 16\), and none of these are equivalent to our width calculation.
As it stands, none of the options A, B, C, or D directly represent \(W\), but \(8(x + 2)\) is equivalent to \(8x + 16\).
Upon further checking options, since any external factors leading to simplifications or transformations might present \(W\) in an unforeseen manner, direct existing values of wfactors do not equate accurately. Please check that provided values are accurate or for any preconditions.