Question

you have 850 feet of fencing to enclose a rectangular plot that borders on a river. if you do not fence the side along the river, find the length and width of the plot that will maximize the area. what is the largest area that can be enclosed?

the width, labeled x in the figure, is \\(\square\\) feet.
(type an integer or decimal.)

Explanation:

Step1: Set up the equations

Let the width be \(x\) (the two sides perpendicular to the river) and the length be \(y\) (the side parallel to the river). The perimeter equation (since we don't fence the side along the river) is \(2x + y=850\), so \(y = 850 - 2x\). The area formula \(A=xy\). Substitute \(y\) into the area formula: \(A(x)=x(850 - 2x)=850x-2x^{2}\).

Step2: Find the vertex of the quadratic function

For a quadratic function \(A(x)=ax^{2}+bx + c\) (here \(a=-2\), \(b = 850\), \(c = 0\)), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
Substitute \(a=-2\) and \(b = 850\) into the formula: \(x=-\frac{850}{2\times(-2)}=\frac{850}{4}=212.5\).

Answer:

\(212.5\)