Advanced maths anyone?

Question

A man has a stone wall alongside a field. He has 1200ft of fencing material and wishes to make a rectangular pen, using the wall as one side.
What should the dimensions of the pen be in order to enclose the largest possible area?
HINT: Area = Length x Width

Answer 1

A = L x W

from the problem, 2 lengths and 1 width have to add up to 1200
2L + W = 1200, W = 1200 - 2L

sub W back into original equation
A = L X (1200 - 2L)
A = 1200L - 2(L^2)

maximum area occurs when the derivative of A is 0
A' = 1200 - 4L = 0
L = 300ft
W = 1200 - 2L = 1200 - 600 = 600ft

I hope thats right lol

Answer 2

We're given that the man has 1200 feet of fencing material. We're also given he wishes to make a rectangular pen, using one side of the wall (hence excluding the fencing)

The perimeter of the rectangular pen would be calculated as follows:

P = (length of the three sides)

Let's call the length of the pen L and the width W. It follows that, since we have three sides,

P = L + L + W
P = 2L + W

However, we know that P = 1200, so

1200 = 2L + W, and solving this equation for W, we get

W = 1200 - 2L

The area of the rectangular pen would be

A = LW

However, we know that W = 1200 - 2L, so

A = L(1200 - 2L), or
A = 1200L - 2L^2

Now that we have this in terms of one variable, let's call this our function, A(L).

A(L) = 1200L - 2L^2

In order to maximize our area, we take the derivative of this function, and then set it to 0.

A'(L) = 1200 - 4L

Setting A'(L) = 0, we have

0 = 1200 - 4L
4L = 1200
L = 300

Therefore, the maximum area occurs when L = 300. Seeing as we want the dimensions, all we need now is the width, but we know that W = 1200 - 2L, so W = 1200 - 2(300) = 1200 - 600
W = 600

Therefore, the dimensions of the pen to enclose the largest possible area would be 300 x 600.

If we wanted, for whatever reason, to *know* what the area would be, we just plug in A(300).

A(300) = 1200(300) - 2(300)^2
A(300) = 360000 - 2(90000)
A(300) = 360000 - 180000 = 180000 square feet.

Answer 3

This cannot be calculated without knowing the length of the wall.

The perimeter is NOT 1200 feet just because you have 1200 feet of fencing material.. The perimeter will be 1200 plus the length of the wall so we must know the length of the wall.

Answer 4

If the wall used in the enclosure is 600 feet long then the area will be 300x600=180,000 square feet of about ten acres.

Answer 5

let width be m and length be n, area be A.

2m+2n=1200
2[m+n}=1200
m+n=1200/2
m+n=600
m=600-n

A=mn
A=n[600-n]
A=600n-n^2
dA/dn=600-2n

Area is maksimum when dA/dn=0

600-2n=0
-2n=-600
n=-600/-2
n=300

m=600-n
m=600-300
m=300

Therefore the dimension of pen should be 300ft * 300ft