The owner of a ranch has 2 , 100 2,100 yards of fencing material which to enclose a rectangular piece of grazing land along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the pasture having the largest area

x  = 1050 yd

y  = 525  yd

A(max)  =  551250 yd²

Step-by-step explanation:

For enclosing a rectangular area (only three sides, since one side will be the river) we have 2100 yards, then the length of fencing material is:

L  =  2100 = x + 2y     ⇒  y  =  ( 2100  -  x  ) / 2

Where x and y are the sides of the rectangle ( x is the parallel side to the river)

The area of the rectangle is:

A = x*y

And as       y = (2100 - x ) / 2

We can express  A as a function of x, getting:

A(x)  =  x* (2100  - x ) /2      or

A(x)  =( 2100*x  - x² )/ 2      ⇒ A(x)  =  1050*x  - (1/2)*x²      (1)

Taking derivatives on both sides of the equation we have

A´(x)  =  1050 - x

A´(x)  =  0   means       1050 - x  = 0

x  =  1050 yd

And as A´´(x)  =  - 1      A´´(x) < 0

We have a maximum for the function at the point x = 1050

Now  

y  = ( 2100  -  x ) /2       then

y  = ( 2100 -  1050 ) / 2

y  = 525 yd

And

A(max)  =  1050* 525

A(max) =  551250 yd²

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step-by-step explanation: