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?

Answer 1

To find the constraint that represents the area of the rectangular garden, we start by using the formula for the perimeter of a rectangle and the formula for the area.

1. **Perimeter Constraint**: The perimeter \( P \) of the rectangle is given by the formula
\[
P = 2 \times (\text{length} + \text{width}) = 2(x + w)
\]
where \( x \) is the length and \( w \) is the width. Given that the total amount of fencing available is 500 feet, we have:
\[
2(x + w) = 500
\]
Simplifying this gives:
\[
x + w = 250 \quad \text{(constraint from fencing)}
\]
Therefore, we can express the width in terms of the length:
\[
w = 250 - x
\]

2. **Area Calculation**: The area \( A \) of the garden is given by:
\[
A = x \times w
\]
Substituting the expression for width from the perimeter constraint, we have:
\[
A = x(250 - x)
\]

So the constraint for the area \( A \) can be expressed as:
\[
A = 250x - x^2
\]
This represents the area of the rectangular garden in terms of its length \( x \).

Thus, the area constraint is:
\[
A = x(250 - x)
\] (with \( A \) representing the area in square feet).