Question

Question #4d05d

Answer 1

The farmer should use length of `212.5`ft , width of `212.5` ft to get maximum area of `45156.25` sq.ft.

Explanation:

Let the length and width of the enclosed area be `l and w` ft

`2l+2w =850 or l+w= 850/2=425.0 :. w = 425-l`

Area of the enclosed portion is `A=l*w=l(425-l)` or

`A= -l^2+425*l = -(l^2- 425l) or -(l^2-425l+212.5^2)+212.5^2`

or `-(l-212.5)^2+45156.25^2` . The vertex form of equation of

parabola is `y=a(x-h)^2+k ; (h.k) ;` being vertex.

Here `h=212.5 and k = 45156.25 , a=-1` . Vertex is at

`(212.5,45156.25)` Since `a` is negative,

parabola opens downward. Therefore vertex is the

maximum point `(212.5, 45156.25) :.` Maximum value is

`45156.25` . `:. l=212.5 ft ; w = (425-212.5)=212.5 ft`.

`A_max=45156.25` sq.ft

The farmer should use length of `212.5`ft , width of `212.5` ft

to get maximum area of `45156.25` sq.ft [Ans]

Answer 2

`212 1/2 xx 141 2/3` ft.

Explanation:

Using the Length (`L`) and Width (`W`) as in the image:

We have the total length of fencing (given as 850 ft):
`3W+2L=850`
`color(white)("xxxxxx")rArr L=1/2(850-3W)`

`"Area"_square=W xx L`

`color(white)("xxxxx")=1/2(850W-3W^2)`

The maximum value will occur when the derivative of this area expression is equal to zero.

`(d" Area")/(d" " W)=1/2(850-6W) =0`

`color(white)("xxxx")rArr W=850/6=141 2/3`

and since `L=1/2(850-3W)`

`color(white)("xxxx")=212 1/2`

Answer 3

Dimention of each corral for maximum possible area of `15502.08` is `106.25` feet by `141.67` feet.

Explanation:

If fencing is required in boundary of two rectangular corrals:

Let the length and width of two individual corals be `x/2 and y`

for each. Then total perimeter of the fencing is `2x+3y=850`

`y= (850-2x)/3 :.` Total enclosed area is `A=x*y` or

`A=x*((850-2x)/3)= -(2x^2)/3+850/3x` . Comparing with

standard quadratic equation `ax^2+bx+c=0` we get here

`a=-2/3 , b= 850/3 , c=0` Discriminant `D=b^2-4ac` or

`D=(850/3)^2-4*(-2/3)*0 ~~80277.78` . Since `a` is

negative maximum value of `A_(max)=-D/(4a)` at `x=-b/(2a)`

Maximum area is `A_(max)~~ (-80277.78)/(4*(-2/3))=30104.17`

sq feet at `x=-850/3/(2*(-2/3)) ~~ 212.5`

`:. x/2=212.5/2=106.25` feet.

`y=(850-2x)/3 = 850-(2*212.5)/3~~141.67` feet

Therefore Dimention of each corral for maximum possible

area of `15502.08` is `x=106.25 , y~~141.67` feet [Ans]