What is the largest rectangular area that can be enclosed with 400 feet of fencing What are the dimensions of the rectangle -- Quadratic Modeling?

Find the dimensions of the rectangular field of maximum area which can be enclosed with 400 feet of fence.

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Let w = width of field

then

(400-2w)/2 = length of field

(200-w) = length of field

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area = w(200-w)

area = 200w-w^2

area = -w^2+200w

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Looking at the coefficient for the w^2 term, we see that it is negative. This indicates that the parabola opens downward and finding the vertex will give you the "maximum".

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Standard vertex form is:

y= a(x-h)^2+k

where

(h,k) is the vertex

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Convert our equation into that form by "completing the square":

area = -w^2+200w

area = -(w^2-200w)

area = -(w^2-200w+10000) + 10000

area = -(w-100)^2 + 10000

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From the above, we see that the vertex is:

(h,k) = (100, 10000)

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This says that when the width=100 feet, the area will be maximized at 10000 square feet.

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Solution:

width = 100 feet

length = 200-w = 200-100 = 100 feet