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

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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

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Answer:x  = 1050 ydy  = 525  ydA(max)  =  551250 yd²Step-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  ) / 2Where 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 ) / 2We can express  A as a function of x, getting:A(x)  =  x* (2100  - x ) /2      orA(x)  =( 2100*x  - x² )/ 2      ⇒ A(x)  =  1050*x  - (1/2)*x²      (1)Taking derivatives on both sides of the equation we haveA´(x)  =  1050 - xA´(x)  =  0   means       1050 - x  = 0x  =  1050 ydAnd as A´´(x)  =  - 1      A´´(x) < 0We have a maximum for the function at the point x = 1050Now  y  = ( 2100  -  x ) /2       theny  = ( 2100 -  1050 ) / 2y  = 525 ydAndA(max)  =  1050* 525A(max) =  551250 yd²

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