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 212.5 feet.
(type an integer or decimal.)
the length, labeled 850 - 2x in the figure, is 425 feet.
(type an integer or decimal.)
the largest area that can be enclosed is \\(\square\\) square feet.
(type an integer or decimal.)

Explanation:

Step1: Recall the area formula for a rectangle

The area formula for a rectangle is \(A = length\times width\). Given \(width=x\) and \(length = 850 - 2x\), then \(A(x)=x(850 - 2x)=850x-2x^{2}\).

Step2: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\) (\(a=- 2\), \(b = 850\), \(c = 0\)), the \(x\) - coordinate of the vertex is \(x=-\frac{b}{2a}\).
Substitute \(a=-2\) and \(b = 850\) into the formula: \(x=-\frac{850}{2\times(-2)}=\frac{850}{4}=212.5\).
We already know \(x = 212.5\) (width) and \(length=850-2x\). Substitute \(x = 212.5\) into \(length\) formula: \(length=850-2\times212.5=850 - 425=425\).

Step3: Calculate the area

Substitute \(x = 212.5\) and \(length = 425\) into the area formula \(A=length\times width\).
\(A=212.5\times425=(200 + 12.5)\times425=200\times425+12.5\times425=85000+5312.5 = 90312.5\)

Answer:

The largest area that can be enclosed is \(90312.5\) square feet.