9514 1404 393
Answer:
58.5 ft by 39 ft
Step-by-step explanation:
Let x represent the length of the two horizontal segments. Then the three vertical segments will be ...
(234 -2x)/3
The total enclosed area is the product of these dimensions:
A = (x)(234 -2x)/3
A = (2/3)(x)(117 -x)
This is the equation of a downward-opening parabola with zeros at x=0 and x=117. The maximum of the parabola will be on the line of symmetry, halfway between these zeros. The value of x there is ...
x = (0 +117)/2 = 58.5
The lengths of the vertical segments are ...
(2/3)(117 -58.5) = 2/3(58.5) = 39
The dimensions of the region enclosing the maximum area are 58.5 ft by 39 ft. The additional vertical segment is 39 ft.
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<em>Comment on maximum area problems</em>
You may have noticed that the total length of the fence allocated to the long sides (2×58.5 = 117) is half the total length of fence and is equal to the total length of fence allocated to the short sides (3×39 = 117).
This relationship is true in all rectangular fencing problems where the area is being maximized for a given total fence length. It doesn't matter how many partitions there are in either direction: <em>the total of horizontal lengths is equal to the total of vertical lengths</em>.
