SOLUTION: Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 151ft

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Question 958223: Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 151ft
Answer by LinnW(1048) About Me  (Show Source):
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perimeter = 2 * length + 2 * width
set L = length , W = width
perimeter = 2L + 2W
since perimeter is to be 151
151 = 2L + 2W
Solving for L,
151 - 2W = 2L
dividing each side by2
%28151+-+2W%29%2F2+=+2L%2F2
%28151+-+2W%29%2F2+=+L
Since area = L*W
+area+=+%28%28151+-+2W%29%2F2%29+%2A+W+
+area+=+%28151W+-+2W%5E2%29%2F2+
multiplying each side by 2
+2+%2A+area+=+%28151W+-+2W%5E2%29+
set y = 2*area
If we maximize the value of y in the equation
+y+=+%28151W+-+2W%5E2%29+
we will can determine the best dimensions for L and W
rewriting
+y+=+-2W%5E2+%2B+151W+
Using the coordinate values a = -2 and b = 151
the value of W where y is maximum occurs when -b/2a is the value of W.
-b%2F2a+=+-%28151%2F2%28-2%29%29
-b%2F2a+=+151%2F4
area will be maximum when W = 151/4
We previously determined +L+=+%28151-2W%29%2F2+ so substituting we get
+L+=+%28151-2%28151%2F4%29%29%2F2+
+L+=+%28151-%28151%2F2%29%29%2F2+
+L+=+%28151%2F2%29%2F2+
+L+=+151%2F4+
So the maximum area will occur when W = 151/4 and L = 151/4