remark: the fence is clearly 420 yards, not 420420.
check the picture
The 2 adjacent rectangles (corrals) form one larger rectangle.
as shown in the figure, let x be the length of the fences perpendicular to the river, then the length of the side opposite to the river will be 420- 2x (yard)
so we can write the following function, which calculates the area A enclosed by the fence, as a function of x.
A(x)=2x(420-2x)
clearly A is a quadratic function, its graph is a parabola. This parabola looks downwards because of the minus of the term -2x
from A(x)=2x(420-2x) we can find that the x intercepts of the parabola are:
one of the roots is x=0
and
420-2x=0,
2x=420
x=210 is the other root.
The x coordinate of the vertex is the midpoint of (0, 210), that is 210/2=105
f(105)=2*105(420-2*105)=210*(420-210=2110*210=44,100 square yard.
the highest point of the parabola is the largest value the function takes, so the maximal area of the fence.
Answer: max area= 44,100 square yard.
