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

Answer 1

Answer:
`60,000 ft^2`

Explanation:

Given

total of 1200 ft of fence is available

Suppose rectangular pasture has following dimension as shown in figure.

total perimeter

`P=4a+3b`

`1200=4a+3b`

`4a=1200-3b`

`a=300-(3)/(4)b`

Area of rectangle is

`A=2a* b`

`A=2(300-(3)/(4)b)b`

`A=2[300 b-(3)/(4)b^2]`

For maximum area, differentiate A w.r.t b

`\frac{\mathrm{d} A}{\mathrm{d} b}=300-(3)/(2)b=0`

`300=(3)/(2)b`

`b=200 ft`

so
`a=300-150=150\ ft`

so area is

`A=2* 150* 200`

`A=60,000 ft^2`