2 word problems dealing with maximums and minimums.?

Question

I am not quite sure if I have these correct.

1. A farmer has 4000ft. of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?
-My answer for this one is 2,000,000 square ft. Doesn't look right.

2. A farmer with 3000 ft. of fencing wants to enclose a rectangular area and then divide it into four pens with three fences parallel to one pair of sides. What are the dimensions which maximize the area?

Answer 1

I'll do #1 and you can follow the example to do #2
The fence will equal 2*length + width = 4000
Thus W = 4000-2L
The Area is L*W = L *(4000-2L) = 4000L -2L^2
This is a parabola that opens down (a=-2) so the vertex will be a maximum.
The vertex is located at L = -b/2a = -4000/-4 = 1000
Thus W = 4000-2L = 4000-2(1000)=2000
So the maximum Area is (1000)(2000)=2000000.
You are absolutely right!

Answer 2

Problem #1. There is no fence along the river so the fence is divided in three equal parts. 1333' 4" X 1333' 4" = 1,777,772 .9sq. ft.

Problem #2. We have four rectangles, but three adjacent sides. The optimum area would be a square with no real difference where it is split. I would divide 3000' by seven for each side (428' 7") and place the interior fences 143' apart.

Answer 3

I don't know what level of math you have, but here goes.

1) first, form two equations in X and Y (X is the distance on one side and Y is the length of the other 2).

X+2Y=4000
X*Y=A

Then, find Y in terms of X and substitute it into the other equation to have area in terms of X only.

Y=2000-(X/2)

X(2000-X/2)=2000X-X^2/2= A

Then take the derivative of this equation and set it equal to 0. This will give you an X value. Plug that X value into the first equation to find your Y value and those are the dimensions.

2)
Very similar to the first problem, but the first equation is different.

2X+5Y=3000
X*Y=A

follow the same steps as in Problem 1 and you'll have your answer.