Question 332820
Perimeter of the rectangular garden = 2*W+L  (one side is covered by the barn)
The area of the garden = L*W
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so we know we have a total of 100 feet of fencing 
therefore,  {{{100 =2*W+L}}}
solve this for L:  {{{L=100-2*W}}}
substitute the L into the area formula :  A=L*W
{{{A=L*W=(100-2*W)*W}}}
{{{A= 100*W -2*W^2}}}
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To find the minimum or maximum area, take the derivative and set to 0
why? because a min or max will occur when the slope is 0, flat part of the function.
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dA/dW = 100 -4*w =0,  solve for W

W=100/4=25   is this a min or max?  if second derivative at  w=25  is negative then its a maximum, why?
because the second derivative relates to the curvature of the function, so if its negative its curving downward
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d^2A/dW^2= -4 thus function is curving downward,  W=25 is a maximum

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now W=25 means that L=100-2*W=50
therefore

max Area=L*W=25*50=1250 ft^2