What is the greatest rectangular area that the farmer

Ответ:

f13vsdbd

14.03.2022

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

$\Rightarrow 2(x+y)=100$

$\Rightarrow x+y=50$

$\Rightarrow y=50-x$

Then the area of the rectangular field will be

As we have to find the maximum area for which the perimeter is 100, so we have to maximize the area function.

$\Rightarrow f(x)=xy$

Putting the value of y,

$\Rightarrow f(x)=x(50-x)$

$\Rightarrow f(x)=50x-x^2$

Taking the derivative on both sides,

$\Rightarrow f'(x)=50-2x$

$\Rightarrow f''(x)=-2$

Since f"(x) is in negative, so the critical value will yield maximum value of the function.

So,

$\Rightarrow f'(x)=0$

$\Rightarrow 50-2x=0$

$\Rightarrow 2x=50$

$\Rightarrow x=25$

So for the value of x, f(x) will have maximum value.

Hence,

$f(25)=25(50-25)=25\times 25=625$

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