what is The greatest rectangular area that the farmer can enclose with 100 m of fencing is m2.

Answer:The greatest rectangular area that the farmer can enclose with 100 m of fencing is 625 m²Step-by-step explanation:The perimeter of the rectangular field is given as 100 m.Let us assume x be the length and y be the breadth of the rectangular field, so[tex]\Rightarrow 2(x+y)=100[/tex][tex]\Rightarrow x+y=50[/tex][tex]\Rightarrow y=50-x[/tex]Then the area of the rectangular field will be [tex]xy[/tex]As we have to find the maximum area for which the perimeter is 100, so we have to maximize the area function.[tex]\Rightarrow f(x)=xy[/tex]Putting the value of y,[tex]\Rightarrow f(x)=x(50-x)[/tex][tex]\Rightarrow f(x)=50x-x^2[/tex]Taking the derivative on both sides,[tex]\Rightarrow f'(x)=50-2x[/tex][tex]\Rightarrow f''(x)=-2[/tex]Since f"(x) is in negative, so the critical value will yield maximum value of the function.So,[tex]\Rightarrow f'(x)=0[/tex][tex]\Rightarrow 50-2x=0[/tex][tex]\Rightarrow 2x=50[/tex][tex]\Rightarrow x=25[/tex]So for the value of x, f(x) will have maximum value.Hence,[tex]f(25)=25(50-25)=25\times 25=625[/tex]