Question

a farmer has 6000 meters of fencing, and wants to enclose a rectangular plot that borders on a river. if the farmer does not fence the side along the river, what is the largest area that can be enclosed? (see the figure.) the largest area that can be enclosed is

Explanation:

Step1: Write the area formula

The area \(A\) of a rectangle is \(A = x(6000 - 2x)=- 2x^{2}+6000x\), where \(x\) is the width and \(6000 - 2x\) is the length.

Step2: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\) (\(a=-2\), \(b = 6000\)), the \(x\) - coordinate of the vertex is \(x=-\frac{b}{2a}\).
Substitute \(a=-2\) and \(b = 6000\) into \(x =-\frac{b}{2a}\), we get \(x=-\frac{6000}{2\times(-2)} = 1500\).

Step3: Calculate the maximum area

Substitute \(x = 1500\) into the area formula \(A=-2x^{2}+6000x\).
\(A=-2\times(1500)^{2}+6000\times1500\)
\(A=-2\times2250000 + 9000000\)
\(A=-4500000+9000000\)
\(A = 4500000\)

Answer:

\(4500000\) square meters