Afarmer will build a rectangular pen for his sheep. a wall will form one side of the pen. the farmer has 28 m of fencing to form the other three sides. the farmer plans to build the pen so that it has its maximum possible area. what will be the dimensions of the farmer’s sheep pen? enter your answers in the boxes.

Answer with explanation:

→The farmer want to build a rectangular pen for his sheep.

→A wall will form one side of the pen.

→The farmer has 28 m of fencing to form the other three sides.

Let length of rectangle = x m

and, Breadth of rectangle = y m

Perimeter of rectangle = ( 2 x + 2 y) meter

Let the side through which wall has been built is along breadth.then equation can be written as

→ 2 x+ 2 y= 28+y

2 x+y=28

y=28-2 x(1)

Area of rectangle = A=Length ×Breadth

A= x y

A= x×(28-2 x)

A= 28 x -2 x²

Differentiating with respect to x, we get

A'=28 -4 x

For Maximal or Minimal value,

A'=0

→28 - 4 x=0

4 x= 28

→→Dividing both side by ,4, we get

x=7

putting the value of x , in equation 1

→y=2 8-2×7

→y=28-14

→y=14

So, length= 7 m

Breadth = 1 4 m

if wall has been side opposite to length, then Dimension of rectangle will be,then equation can be written as

 2 x + 2 y= 28+x

x+2 y=28

→x=28-2 y

Applying same method done above,we will get

Length= 14 m

Breadth = 7 m

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