To determine the equation for the dimensions of Jill's horse paddock, we start by noting that she has 120 feet of fencing and will use the barn as one side of the rectangular paddock.
Let's define the variables:
- Let \( x \) be the width of the paddock (the side perpendicular to the barn).
- Let \( y \) be the length of the paddock (the side parallel to the barn).
Since the barn is used as one side of the paddock, Jill only needs to fence three sides: two widths (2x) and one length (y). We can set up the equation for the total perimeter of the fencing:
\[ 2x + y = 120 \]
Next, we need to incorporate the area of the paddock, which is given as 1,800 square feet. The area \( A \) of a rectangle is calculated as:
\[ A = x \times y \]
Since the area is 1,800 square feet, we can express this as:
\[ x \cdot y = 1800 \]
Now we have a system of two equations:
- \( 2x + y = 120 \)
- \( xy = 1800 \)
To find a single equation in terms of \( x \), we can solve the first equation for \( y \):
\[ y = 120 - 2x \]
Now, substitute this expression for \( y \) into the second equation:
\[ x(120 - 2x) = 1800 \]
Distributing \( x \):
\[ 120x - 2x^2 = 1800 \]
Rearranging gives us a standard form quadratic equation:
\[ 2x^2 - 120x + 1800 = 0 \]
Dividing everything by 2 simplifies it:
\[ x^2 - 60x + 900 = 0 \]
This quadratic equation can be used to find the width \( x \) of the horse paddock. Hence, the relevant equation derived from the initial conditions is:
\[ x^2 - 60x + 900 = 0 \]
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