Question

a farmer wants to fence in three sides of a rectangular field shown below with 960 feet of fencing. The other side of the rectangle will be a river. If the enclosed area is to be a maximum, find the dimensions of the field. ​

Answer 1

9514 1404 393

Answer:

  240 ft by 480 ft

Step-by-step explanation:

Area is maximized when the long side is half the total length of the fence. That makes the short side (out from the river) be half the length of the long side.

The fenced field dimensions are 240 feet by 480 feet.

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You can let x represent the length of the long side. Then the length of the short side is half the remaining fence: (960 -x)/2.

The total area is the product of these dimensions:

  A = x(960 -x)/2

We note that this is the equation of a parabola with zeros at x=0 and x=960. The maximum will be found on the line of symmetry, halfway between the zeros. That is at x = (0 +960)/2 = 480.

The area is maximized for a long-side dimension of 480 feet. The short sides are 240 feet.