Farmer Ed has 700700 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?

Length = 350 m, Width = 175 m, Area = 61250 m^2

Step-by-step explanation:

(Assuming the correct value is 700 meters of fencing)

Ed need to make a rectangle using his fences. The rectangle has 4 sides, and the opposite sides have the same length. So, calling "a" the bigger side and "b" the smaller side, we have that the area of the rectangle is a*b, and its perimeter is 2*a + 2*b

Ed will not fence the side along the river. So, to maximize the area, the side of the retangle that will not be fenced needs to be one of the bigger sides. Now, the perimeter of fencing will be a + 2*b, while the area still is a*b.

Ed has a total of 700 meters of fencing, so we have that a + 2*b = 700 - > a = 700 - 2*b.

Using this value of "a" in the area equation, we have that the area is:

(700 - 2*b) * b = 700*b - 2*b^2

To maximize this area, we derivate this expression and make it equal to zero, then we will find the value of "b" that maximizes this expression:

the derivate of 700*b - 2*b^2 is 700 - 4b. Making it equal to zero, we have:

700 - 4*b = 0 - > b = 175

With b=175, we find "a" with:

a + 2*b = 700 - > a + 350 = 700 - > a = 350

So, the length of the plot to maximize the area is 350 m, the width is 175 m, and the area is 350*175 = 61250 m^2