Question

you have 200 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 feet. (type an integer or decimal.)

Explanation:

Step1: Set up the area function

Let the width be \(x\) (perpendicular to the river) and the length be \(l = 200 - 2x\) (parallel to the river). The area \(A\) of a rectangle is \(A=x\times l\). Substituting \(l = 200 - 2x\) into the area formula, we get \(A(x)=x(200 - 2x)=200x-2x^{2}\).

Step2: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\) (\(a=- 2\), \(b = 200\), \(c = 0\)), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
Substitute \(a=-2\) and \(b = 200\) into the formula: \(x=-\frac{200}{2\times(-2)}=\frac{-200}{-4}=50\).

Answer:

\(50\)