Question
The length of a rectangular frame is 4 inches longer than 3 times the width. Which of the following equations describes the relationship between the width (w) of the frame in inches, and the area (A) in square inches?
Responses
cap A is equal to 4 w open paren w plus 3 close paren
Responses
cap A is equal to 4 w open paren w plus 3 close paren
Answers
GPT-4o mini
To express the relationship between the width \( w \) of the rectangular frame and the area \( A \), we first need to determine the length \( l \) in terms of the width \( w \).
According to the problem, the length \( l \) is 4 inches longer than 3 times the width:
\[
l = 3w + 4
\]
The area \( A \) of a rectangle is given by the formula:
\[
A = l \times w
\]
Substituting the expression for \( l \):
\[
A = (3w + 4) \times w
\]
Now we can distribute \( w \):
\[
A = 3w^2 + 4w
\]
So the equation that describes the relationship between the width \( w \) and the area \( A \) is:
\[
A = 3w^2 + 4w
\]
Therefore, the correct form of the relationship based on the problem statement and the area formula does not match the response you provided, as it indicates \( A \) is equal to \( 4w(w + 3) \), which is not correct.
So, the correct equation is:
\[
A = 3w^2 + 4w
\]
According to the problem, the length \( l \) is 4 inches longer than 3 times the width:
\[
l = 3w + 4
\]
The area \( A \) of a rectangle is given by the formula:
\[
A = l \times w
\]
Substituting the expression for \( l \):
\[
A = (3w + 4) \times w
\]
Now we can distribute \( w \):
\[
A = 3w^2 + 4w
\]
So the equation that describes the relationship between the width \( w \) and the area \( A \) is:
\[
A = 3w^2 + 4w
\]
Therefore, the correct form of the relationship based on the problem statement and the area formula does not match the response you provided, as it indicates \( A \) is equal to \( 4w(w + 3) \), which is not correct.
So, the correct equation is:
\[
A = 3w^2 + 4w
\]