Farmer Ed has 600 meters of fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does 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?

Let f(x) be a real function and f'(x) its derivative. The values for which f'(x)=0 are called the critical points. If x=a is a critical point of f(x) and

f''(a)>0, then x=a is a minimum

f''(a)<0, then x=a is a maximum

The question requires us to maximize the area of the rectangular plot, knowing the perimeter of the fencing is 600 m.

The perimeter of a rectangle of sides x and y is

[tex]P=2x+2y[/tex]

Since the side along the river (assumed to be y) won't be fenced, then the perimeter of the fence is

[tex]P=2x+y=600[/tex]

Solving for y

[tex]y=600-2x[/tex]

Now, the area of the rectangle is

[tex]A=x.y[/tex]

Using the relation found above:

[tex]A=x(600-2x)=600x-2x^2[/tex]

Find the derivative

[tex]A'=600-4x[/tex]

Equal to zero

[tex]600-4x=0[/tex]

Solve for x

[tex]x=150\ m[/tex]

Find the second derivative

[tex]A''=-4[/tex]

Since it's negative, the value x=150 is a maximum

The value of y is

[tex]y=600-2\cdot 150=300\ m[/tex]

Thus, the dimensions of the plot are x=150 m, y=300 m

And the maximum area of the plot is

[tex]A=150\cdot 300=45,000\ m^2[/tex]

Farmer Ed has 600 meters of​ fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does 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?

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Farmer Ed has 600 meters of fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does 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?