Question

a garden is to be laid out in a rectangular area and protected by a chicken wire fence. What is the largest possible area of the garden if only 100 running feet of chicken wire is available for the fence?

Answer 1

The largest possible area for a given amount of material enclosing a rectangle will always be a perfect square with each side being one quarter of the amount of material.

The proof is:

M=2x+2y, solve for y

y=(M-2x)/2

A=xy, and using y from above...

A=(Mx-2x^2)/2

dA/dx=(M-4x)/2

d2A/dx2=-2

Since acceleration is always negative, when velocity is zero, it will be at an absolute maximum for A(x)

dA/dx=0 when M-4x=0, 4x=M, x=M/4

So the maximum area occurs when x=M/4, and from earlier:

y=(M-2x)/2  then use x=M/4 in this and get:

y=(M-M/2)/2

y=(2M-M)/4=M/4  so x=y=M/4, a perfect square.

So for your particular example...

M=2x+2y and M=100

100=2x+2y

50=x+y

y=50-x

Now:

A=xy, using y found above...

A=50x-x^2

dA/dx=50-2x

d2A/dx2=-2  so an absolute maximum occurs when dA/dx=0

dA/dx=0 when 50-2x=0, 2x=50, x=25...

and from earlier, y=50-x, using x from above y=50-25=25

so y=x=25

So the maximum area with 100 feet of fencing is 25^2=625ft^2

Answer 2

I believe that 25*25=625 and that is the biggest area