Answer:[tex]60,000 ft^2[/tex]
Step-by-step explanation:
Given
total of 1200 ft of fence is available
Suppose 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]
