Question
For a rectangular pool with a fixed area, the length of the pool varies inversely with the width. Which equation represents this situation for an area of 512 square feet?
Answers
GPT-4o mini
To represent the situation where the length of a rectangular pool varies inversely with its width, we start with the relationship between area, length, and width. The area \( A \) of the rectangle is given by:
\[
A = \text{length} \times \text{width}
\]
In this case, we are given that the area is fixed at 512 square feet, so we have:
\[
lw = 512
\]
where \( l \) is the length and \( w \) is the width of the pool.
Since the length varies inversely with width, we can express the length as:
\[
l = \frac{k}{w}
\]
for some constant \( k \). To determine \( k \), we can use the area equation \( lw = 512 \). Substituting \( l \) in terms of \( w \):
\[
\left(\frac{k}{w}\right)w = 512
\]
Simplifying this gives us:
\[
k = 512
\]
Now, substituting \( k \) back into the length equation gives us:
\[
l = \frac{512}{w}
\]
Thus, the equation representing the relationship between the length \( l \) and the width \( w \) of the rectangular pool with a fixed area of 512 square feet is:
\[
l = \frac{512}{w}
\]
\[
A = \text{length} \times \text{width}
\]
In this case, we are given that the area is fixed at 512 square feet, so we have:
\[
lw = 512
\]
where \( l \) is the length and \( w \) is the width of the pool.
Since the length varies inversely with width, we can express the length as:
\[
l = \frac{k}{w}
\]
for some constant \( k \). To determine \( k \), we can use the area equation \( lw = 512 \). Substituting \( l \) in terms of \( w \):
\[
\left(\frac{k}{w}\right)w = 512
\]
Simplifying this gives us:
\[
k = 512
\]
Now, substituting \( k \) back into the length equation gives us:
\[
l = \frac{512}{w}
\]
Thus, the equation representing the relationship between the length \( l \) and the width \( w \) of the rectangular pool with a fixed area of 512 square feet is:
\[
l = \frac{512}{w}
\]