Question

A rectangular garden is to be enclosed with 500 feet of fencing. The garden has a length of x feet and an area A square feet.

Which constraint represents the area?
A= x^2 + 250x
A= -x^2 + 250x

Answer 1

To find the correct constraint representing the area \( A \) of the rectangular garden, we need to use the information given about the perimeter and the dimensions of the garden.

We know that the perimeter \( P \) of a rectangle is given by the formula:

\[
P = 2L + 2W
\]

where \( L \) is the length and \( W \) is the width. According to the problem, the total amount of fencing available is 500 feet, so we can express the perimeter as:

\[
2L + 2W = 500
\]

Dividing by 2 gives:

\[
L + W = 250
\]

If we let the length \( L \) be \( x \), then the width \( W \) can be expressed as:

\[
W = 250 - x
\]

The area \( A \) of the rectangle is given by:

\[
A = L \times W
\]

Substituting \( L = x \) and \( W = 250 - x \) into the area formula gives us:

\[
A = x(250 - x)
\]

Expanding this:

\[
A = 250x - x^2
\]

This simplifies to:

\[
A = -x^2 + 250x
\]

Therefore, the correct constraint that represents the area is:

\[
A = -x^2 + 250x
\]

So the answer is:

**\( A = -x^2 + 250x \)**.