Question 724271: I need help setting a word problem up. If someone could please show me the equations I need to solve this:
a farmer is fencing a rectangular area for cattle using a straight portion of the river as one side of the rectangle. There is no fencing along the river. If the farmer has 1200 feet of fence, find the dimensions for the rectangular area that gives a maximum area for the cattle.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! a farmer is fencing a rectangular area for cattle using a straight portion of the river as one side of the rectangle. There is no fencing along the river. If the farmer has 1200 feet of fence, find the dimensions for the rectangular area that gives a maximum area for the cattle.
***
let x=width (perpendicular to river)
let y=length (parallel to river)
2x+y=1200
y=1200-2x
area=length*width
xy=x(1200-2x)=1200x-2x^2
Area=-2x^2+1200x
complete the square
Area=-2(x^2-600x+300^2)+2(300^2)
Area=-2(x-300)^2+180000
This is an equation of the standard form for a parabola that opens downward: y=-A(x-h)^2+k, (h,k)=(x,y) coordinates of the vertex from which the x-value which gives the maximum area and the maximum area itself is obtained.
x=300
y=1200-2x=600
dimensions for the rectangular area that gives a maximum area for the cattle:
width=300 ft
length=600 ft
maximum area=180,000 sq ft
|
|
|
