Answer:
The greatest rectangular area that the farmer can enclose with 100 m of fencing is 625 m²
Step-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]
