Question 535681
Be careful. This could be a "trick" question. If it is required that the field be a rectangular shape, the maximum area would be enclosed by a field that is square. In that case the field would have all four sides be of length 25 meters. This would give you a perimeter of 100 meters and would enclose a field of dimensions 25 meters by 25 meters, resulting in an enclosed area of 25 times 25 = 625 square meters.
.
But if the farmer is allowed to build a circular fenced area, this will result in a different area being enclosed. The 100 meters of fence would be the circumference of the field. Since we know that the formula for circumference of a circle is:
.
{{{C = 2*pi*R}}}
.
Where C represents the circumference (in this case C = 100 meters) and R is the radius of the circular field. We can solve for the radius of this field by dividing both sides of the equation by {{{2*pi}}} to get:
.
{{{R = C/(2*pi)}}}
.
And substituting 100 for C the equation becomes:
.
{{{R = 100/(2*pi)}}}
.
Dividing the numerator of 100 by {{{2*pi}}} results in finding that the radius R is 15.91549431 meters.
.
We also know from geometry that the Area of a circle is given by the formula:
.
{{{A = pi*R^2}}}
.
in which A represents the area and R again represents the radius. We know that for this problem R equals 15.91549431 meters. Substitute this value into the equation for the area and you get:
.
{{{A = pi*(15.91549431)^2}}}
.
Square the radius and multiply that result by {{{pi}}} and you find that the circular field contains an area of 795.7747154 square meters. 
.
This is more area than if the field were a square of 25 meters by 25 meters. The area made by fencing a circle from 100 meters of fence contains nearly 171 square meters more (795.7747154 square meters minus 625 square meters = 170.7747154 square meters). 
.
Hope this helps you understand the problem a little better.
.