A farmer has 1700 feet of fence with which to fence a rectangular plot of land!

Question

A farmer is fencing a rectangular area for cattle and uses a straight portion of a river as one side of the rectangle, as illustrated in the figure. Note that there is no fence along the river. If the farmer has 1700 feet of fence, find the dimensions for the rectangular area that gives the maximum area for the cattle.

 

The width of the rectangular is how many ft?

The length of the rectangular is how many ft? 

Answer 1

A square gives the greatest area. If the river were not there, you would divide 1700 by 4 to get a square 425 feet by 425 feet. Because of the river, add 425 feet to the side opposite the river to get a rectangle 425 feet by 850 feet.

Now for the algebra that you need.

P=2l+2w but there is only one length.

P=l+2w

1700=l+2w

1700-2w=l

A=lw

A=(1700-2w)w

A=1700w-2w²

This is a parabola that opens downward so there is a maximum point, the vertex, which is (h,k).

h=-b/2a is the "maximizing point" and k is the maximum area

-b/2a=-1700/(-4)

-b/2a=425 feet

One width will be 425 feet, the length will be (425+425)=850 feet, and the second width will also be 425 feet.

A=lw

The maximum area will be A=850*425=361,250 square feet.

Notice that if you substitute 425 into the equation A=1700w-2w² you will get the same answer.

1700*425-2*425²

722,500-2*180,625

722,500-361,250

 

Answer 2

Hi Jackie.  If you think of the dimensions and try different combinations that make up a long length and 2x a short length to add up to 1700.  either 800x450 or 900x400 will give you 360,000 sqft.  Any shorter start taking away from the maximum square area.