Questions: A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 1200 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

Solution Steps

Given a fixed length of wire \(L = 1200\) meters, find the dimensions of a rectangular plot of farmland that maximizes the area, with one side along a river and the other three sides fenced.

Let the length of the side parallel to the river be \(x\) meters. Then, the other two sides will each have a length of \(\dfrac{L - x}{2}\) meters. The area \(A\) can be expressed as \(A(x) = x \cdot \dfrac{L - x}{2}\).

The derivative of \(A(x)\) with respect to \(x\) is \(\dfrac{dA}{dx} = \dfrac{L}{2} - x\). Setting this equal to zero gives \(x = \dfrac{L}{2}\).

The second derivative of \(A(x)\) is negative, indicating that the critical point \(x = \dfrac{L}{2}\) is indeed a maximum.

The dimensions that maximize the area are: length \(x = 600\) meters, and width \(\dfrac{L - x}{2} = 300\) meters.

The maximum area is \(A_{max} = \dfrac{L^2}{8} = 180000\) square meters.

Final Answer: