You have 800 feet of fencing and you want to make two fenced in enclosures by splitting one enclosure in half. what are the largest dimensions of this enclosure that you could build?

You have 800 feet of fencing and you want to make two fenced in enclosures by splitting one enclosure in half. What are the largest dimensions of this enclosure that you could build?

Make a drawing and denote:

x = half of the length of the enclosure

2x = the length of the enclosure

y = the width of the enclosure

P = 800 ft the perimeter

The perimeter of the two enclosures can be expressed P = 4x + 2y thus

4x + 3y = 800

Solving for y:

click here to see all the equation solution steps

y = 800/3 - 4x/3

The area of the two enclosure is A = 2xy.

Substituting y = 800/3 - 4x/3 in A = 2xy we get

A = 2x(800/3 - 4x/3)

A =1600x/3 - 8x^2/3

We need to find the x for which the parabolic function A = (- 8/3)x^2 + (1600/3)x has maximum: 

x max = -b/2a, a = (-8/3), b = 1600/3

x max = (-1600/3)/(2*(-8/3))

x max = 100 ft

y = 800/3 - 4*100/3

y = 133.33 ft

2x = 2*100

2x = 200 ft

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