Question

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 2500 m of wire at your​ disposal, what is the largest area you can​ enclose, and what are its​ dimensions?

Answer 1

Answer:

$A = 781250\,m^{2}$, $x = 625\,m$, $y = 1250\,m$

Step-by-step explanation:

The perimeter covered by the electric fence in meters is:

$2\cdot x + y = 2500$

The area of the rectangle is:

$A = x\cdot y$

$A = x \cdot (2500-2\cdot x)$

Let differentiate the previous equation and equates to zero:

$2500-4\cdot x = 0$

The critical point is:

$x = 625\,m$

By the Second Derivative Text, it is proved that critical point lead to a maximum:

$\frac{d^{2}A}{dx^{2}} = -4$

The other side of the rectangle is:

$y = 1250\,m$

The largest area than can be enclosed is:

$A = (625\,m)\cdot (1250\,m)$

$A = 781250\,m^{2}$

The dimensions of the triangle are:

$x = 625\,m$

$y = 1250\,m$