Maximum Area

Question

You have 32 feet of fencing available to enclose a rectangular area bordering a river for a family of ducks. If no fencing is required along the river, find the dimensions of the enclosure that will yield the maximum area. What is the maximum area?

Answer 1

 

 1 length and 2 widths. The other length is the river.

 

 Perimeter = 2 * width + length

 

    32 = 2w + L ---> L = 32  -  2w

 

  Area = length x width

 

  Area = (32 - 2w)w

 

  So the area function in terms of the width is A(w) = (32 - 2w)w = (16 - w)(2w)

  and it is a parabola.

 

  I prefer finding the vertex by averaging the zeros and then plugging it in.

 

  To find the zeros, set  (16 - w)(2w) = 0

      16 - w = 0 ---> w = 16

       2w = 0 ---> w = 0

 

   The average is (16+0)/2 = 16/2 = 8

 

  The axis of symmetry is x=8.

  A(8) = (16 - 8)(2*8) = 8*16 = 128

 

So the max area is 128, which occurs when the width is 8 ( forcing the length to be 16)

 

The ducks can go into the river and escape the cage ;-)