Suppose the width of the play area is \(x\) feet. Because there are three sections of fence along the width of the play area, that leaves \(300 - 3x\) feet of fence for its length.
The area of the play area should be \(6000\) square feet, so we have the equation
\begin{equation*}
x(300 - 3x) = 6000
\end{equation*}
This is a quadratic equation. In standard form,
\begin{align*}
3x^2 - 300x + 6000 \amp= 0\amp\amp \blert{\text{Divide each term by 3.}}\\
x^2 - 100x + 2000 \amp= 0
\end{align*}
The left side cannot be factored, so we use the quadratic formula with \(a = \alert{1}\text{,}\) \(b = \alert{-100}\text{,}\) and \(c = \alert{2000}\text{.}\)
\begin{align*}
x \amp=\frac{-(\alert{-100}) \pm\sqrt{(\alert{-100})^2 - 4(\alert{1})(\alert{2000})}}{2(\alert{1})}\\
\amp= \frac{100 \pm\sqrt{2000}}{2}\approx \frac{100 \pm 44.7}{2}
\end{align*}
When we evalute this last expression, we get two different positive values, \(x = 72.4\) or \(x=27.6\text{.}\) Both values give solutions to the problem. To find the length of the playground in each case, we substitute \(x\) into \(300-3x.\)
If the width of the play area is \(72.4\) feet, the length is \(300 - 3(72.4)\text{,}\) or \(82.8\) feet.
If the width is \(27.6\) feet, the length is \(300 - 3(27.6)\text{,}\) or \(217.2\) feet.
The dimensions of the play area can be 72.4 feet by 82.8 feet, or it can be 27.6 feet by 217.2 feet.