A farmer wishes to enclose a rectangular pasture with 1200 ft of fence. He needs to divide this rectangle into two smaller rectangles by running a fence down the middle of it parallel to the other two sides. He has 1200 ft of fence with which to do this. Find the maximum total area of the enclosure

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Question:

A farmer wishes to enclose a rectangular pasture with 1200 ft of fence. He needs to divide this rectangle into two smaller rectangles by running a fence down the middle of it parallel to the other two sides. He has 1200 ft of fence with which to do this. Find the maximum total area of the enclosure

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Answer:[tex]60,000 ft^2[/tex]Step-by-step explanation:Giventotal of 1200 ft of fence is availableSuppose rectangular pasture has following dimension as shown in figure. total perimeter[tex]P=4a+3b[/tex][tex]1200=4a+3b[/tex][tex]4a=1200-3b[/tex][tex]a=300-\frac{3}{4}b[/tex]Area of rectangle is [tex]A=2a\times b[/tex][tex]A=2(300-\frac{3}{4}b)b[/tex][tex]A=2[300 b-\frac{3}{4}b^2][/tex]For maximum area, differentiate A w.r.t b[tex]\frac{\mathrm{d} A}{\mathrm{d} b}=300-\frac{3}{2}b=0[/tex][tex]300=\frac{3}{2}b[/tex][tex]b=200 ft[/tex]so [tex]a=300-150=150\ ft[/tex]so area is [tex]A=2\times 150\times 200[/tex][tex]A=60,000 ft^2[/tex]

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