Question

A farmer has 180
linear feet of fencing to create a rectangular pen for his goats. The function, A(x)
, represents the area in square feet of the pen where the length of one side, in feet, is x
.
Which function can be used to model A(x)
?

Responses

Answer 1

To model the area \( A(x) \) of a rectangular pen using 180 linear feet of fencing, we need to determine the relationship between the length and width of the rectangle.

Let:
- \( x \) represent the length of one side of the rectangular pen.
- \( y \) represent the width of the pen.

The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2x + 2y
\]
Given that the farmer has 180 linear feet of fencing, we can set up the equation:
\[
2x + 2y = 180
\]
Dividing through by 2 simplifies this to:
\[
x + y = 90
\]
Now, we can express \( y \) in terms of \( x \):
\[
y = 90 - x
\]

The area \( A \) of the rectangle is given by:
\[
A = x \cdot y
\]
Substituting \( y \) in terms of \( x \):
\[
A(x) = x(90 - x)
\]
This can be simplified to:
\[
A(x) = 90x - x^2
\]

Thus, the function that models the area \( A(x) \) is:
\[
A(x) = -x^2 + 90x
\]

This is a quadratic equation that represents the area of the rectangular pen in square feet as a function of the length \( x \).