Given:
The farmer has 500 yards of fencing with which to enclose a rectangular paddock.
Let x be the width of the rectangle and l be the length of the rectangle.
The equation for the total fencing available is given by,
[tex]\begin{gathered} 2x+2l=500 \\ \text{Simplify} \\ x+l=250 \\ l=250-x \end{gathered}[/tex]So, the area is given as,
[tex]\begin{gathered} A=x\cdot l \\ A=x\cdot(250-x) \\ A=250x-x^2 \end{gathered}[/tex]Find the critical values,
[tex]\begin{gathered} A^{\prime}=\frac{d}{dx}(250x-x^2) \\ A^{\prime}=250-2x \\ \text{Set A'=0} \\ 250-2x=0 \\ x=125 \end{gathered}[/tex]Test critical value,
[tex]A^{\doubleprime}=\frac{d^2}{dx^2}(250x-x^2)=-2<0\text{ for all x, }\Rightarrow x=125\text{ is maximum}[/tex]So, the maximum area is,
[tex]A=250x-x^2=250(125)-125^2=15625\text{ square yards.}[/tex]Answer: the maximum area is 15625 square yards.
