Question

Suppose that the owner of a farm has 1,800 feet of fencing available, and they would like to enclose a rectangular portion of the land to allow animals to graze along side a straight portion of a river. The animals don't like the river water, so fencing is not necessary along the river. What is the largest area that they can enclose using the fencing? What are the necessary dimensions (i.e., the length and width) of fencing in order to achieve that area?

Answer 1

(a)
`405000\text{ feet}^2` (b)
`450\text{ feet and }900\text{ feet}`

Explanation:

GIVEN: Suppose that the owner of a farm has
`1,800 \text{ feet}` of fencing available, and they would like to enclose a rectangular portion of the land to allow animals to graze along side a straight portion of a river. The animals don't like the river water, so fencing is not necessary along the river.

TO FIND: What is the largest area that they can enclose using the fencing? What are the necessary dimensions of fencing in order to achieve that area?

SOLUTION:

Let the length and width of area enclosed be
`a\text{ and }b`

as one side of area does not require fencing, total perimeter of rectangular portion.

`2a+b=1800`

`b=1800-2a`

Area of rectangular portion
`=\text{length}*\text{width}`

`=ab`

putting value of
`b` in equation

`area=a(1800-2a)`

`area=1800a-2a^2`

to maximize area
`(d(area))/(da)=0`

`1800-4a=0`

`a=450\text{ feet}`

now,

`b=1800-2a`

`b=900\text{ feet}`

area of rectangular portion
`=450*900=405000\text{ feet}^2`

Hence largest area of enclosure is
`405000\text{ feet}^2` and length and width are
`450\text{ feet and }900\text{ feet}`