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{align*}
\text{width} \times \text{length} \amp = \text{Area}\\
x(300 - 3x) \amp = 6000
\end{align*}
This is a quadratic equation. We use the distributive law and write the 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 of this equation cannot be factored, so we use the quadratic formula with \(a = 1\text{,}\) \(b = -100\text{,}\) and \(c = 2000\text{.}\)
\begin{align*}
x \amp=\frac{-(-100) \pm\sqrt{(-100)^2 - 4(1)(2000)}}{2(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 for the width of the play area, \(x = 72.4\) 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.
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If the width is \(27.6\) feet, the length is \(300 - 3(27.6)\text{,}\) or \(217.2\) feet.