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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Question:

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

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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