Answer:
Length=16 ft
Width=8 ft
Step-by-step explanation:
Let width be w and length will be 128/w since area is product of length and width
Perimeter will be w+w+128/w. Note that along the river, fencing isn't needed that is why perimeter equals 2w+128/w
Getting the first differential of perimeter with respect to w then
[tex]2-\frac {128}{w^{2}}[/tex]
At critical point
[tex]2-\frac {128}{w^{2}}=0[/tex]
[tex]2=\frac {128}{w^{2}}[/tex]
[tex]w^{2}=64\\w=8[/tex]
Therefore, since length is 128/w, length is 128/8=16 ft
Therefore, maximum length and width are 16 ft and 8ft respectively
For least amount of fencing, length and width of rectangular area should be 16 feet and 8 feet respectively.
Let us consider length and width are l and w respectively.
Area of rectangular region = [tex]l * w[/tex]
[tex]l*w = 128\\\\w=\frac{128}{l}[/tex]
Since, no fencing needed along the river.
So, perimeter of rectangular region, P = [tex]2w+l[/tex]
Substituting the value of w in above expression.
We get, [tex]P=2(\frac{128}{l} )+l=\frac{256}{l}+l[/tex]
For least amount of fencing, perimeter should be minimum
Differentiate perimeter expression with respect to length l.
[tex]\frac{dP}{dl} =-\frac{256}{l^{2} }+1=0 \\\\l^{2}=256\\\\l=\sqrt{256}=16feet[/tex]
So, [tex]w=\frac{128}{16} =8feet[/tex]
Thus, For least amount of fencing, length and width of rectangular area should be 16 feet and 8 feet respectively.
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