Respuesta :
Answer:
(a) [tex]405000\text{ feet}^2[/tex] (b) [tex]450\text{ feet and }900\text{ feet}[/tex]
Step-by-step explanation:
GIVEN: Suppose that the owner of a farm has [tex]1,800 \text{ feet}[/tex] 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 [tex]a\text{ and }b[/tex]
as one side of area does not require fencing, total perimeter of rectangular portion.
[tex]2a+b=1800[/tex]
[tex]b=1800-2a[/tex]
Area of rectangular portion [tex]=\text{length}\times\text{width}[/tex]
[tex]=ab[/tex]
putting value of [tex]b[/tex] in equation
[tex]area=a(1800-2a)[/tex]
[tex]area=1800a-2a^2[/tex]
to maximize area [tex]\frac{d(area)}{da}=0[/tex]
[tex]1800-4a=0[/tex]
[tex]a=450\text{ feet}[/tex]
now,
[tex]b=1800-2a[/tex]
[tex]b=900\text{ feet}[/tex]
area of rectangular portion [tex]=450\times900=405000\text{ feet}^2[/tex]
Hence largest area of enclosure is [tex]405000\text{ feet}^2[/tex] and length and width are [tex]450\text{ feet and }900\text{ feet}[/tex]
