Respuesta :
Answer:
1700 ft
Step-by-step explanation:
Let the dimensions of the plot of land be x any y
Where:
- x is fenced using heavy-duty fencing selling for $3 per foot
- y is fenced using standard fencing costing $2 per foot.
Perimeter of the Land=2x+2y
Cost of Fencing =$3(2x)+$2(2y)=6x+4y
The budget that will be entirely used for fencing is $6000.
- Therefore: 6x+4y=6000
- Make x the subject
- [tex]x=\dfrac{6000-4y}{6}[/tex]
Area of the Land, A(x,y)=xy
Substitute [tex]x=\dfrac{6000-4y}{6}[/tex] into A(x,y)
[tex]A(y)=\dfrac{6000y-4y^2}{6}\\A'=\dfrac{3000-4y}{3}[/tex]
The Area is maximized when its derivative equals zero.
3000-4y=0
4y=3000
y=750
Recall: [tex]x=\dfrac{6000-4y}{6}=\dfrac{6000-4(750)}{6}=\dfrac{6000-3000}{6}=500[/tex]
Therefore: x=500ft, y=750ft
Since side y is fenced using standard fencing,
Feet of standard Fencing required = 2y=2 X 750 =1500 ft
So, the required length of the sides that have standard fencing is 1500 ft.
Area of the rectangle:
The area of rectangle is the region covered by the rectangle in a two-dimensional plane. A rectangle is a type of quadrilateral, a 2d shape that has four sides and four vertices.
Let [tex]x[/tex] ft be the length of the sides that duty fencing and [tex]y[/tex] ft be the length of the sides that have standard fencing.
So, the area will be calculated by the above formula we get,
[tex]Area=xy[/tex]...(1)
Now, the cost of fencing is,
[tex]3x+2y=6000\\y=3000-\frac{3x}{2}[/tex]...(2)
Now, substituting equation (2) in equation (1) we get,
[tex]A=3000x-\frac{3x^2}{2}[/tex]
Now, differentiating the above equation we get,
[tex]\frac{dA}{dx} =3000-3x=0\\x=1000[/tex]
Substituting [tex]x=1000 ft[/tex] in equation (2) we get,
[tex]y=3000-\frac{3\times 1000}{2}\\=1500 ft[/tex]
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