Question 1163079
you have a total of 180 meters of fence and there is a pre-existing fence on one side.


if i assume that the 180 meters of fence is the length of the existing fence that will be taken down and used to enclose the area, then this means your perimeter will be equal to 180 meters.
the largest area is when the area is a square, so you need to do the following.
s = the length of a side of the enclosed area.
the perimeter is equal to 4 * s
perimeter = 4 * s
since perimeter = 180, then you get 180 = 4 * s
solve for s to get s = 45
that would be the length of a side of the enclosed area.
the area would be s^2 = 2025 square meters.


if i assume that the 180 meters of fence is new fence that will be attached to the old fence to make the area, then i have to assume there is sufficient length of the old fence to make one side of the enclosure, whatever that length needs to be.
the 180 of new fence will surround 3 sides of the enclosure.
that means 60 meters on a side (180 / 3 = 60)
the fourth side will be 60 meters of the old fence.
the area would then be 60^2 = 3600 square meters.


bottom line:
if the 180 meters is the length of the existing fence, then your maximum area will be 45^2 = 2025 square meters.
if the 180 meters is the length of new fence that will be attached to old fence, then your maximum area will be 60^2 = 3600 square meters.


to convince yourself that a square gives you the maximum area, do the following:
assume total of length and width = 10
possible integer values of length and width and area = length * width would be:
1 * 9 = 9
2 * 8 = 16
3 * 7 = 21
4 * 6 = 24
5 * 5 = 25 ***** maximum area when length = width
6 * 4 = 24
7 * 3 = 21
8 * 2 = 16
9 * 1 = 9


if you were told the length of the existing fence, then the problem would need to be calculated differently.


i think the maximum enclosed area would still be a square, but let's see if this makes sense.
if the existing fence is at least 60 meters, then the above second assumption applies.


assume the existing fence is 30 meters.
now you have a total perimeter of 30 + 180 = 210 meters.
divide that by 4 to get s = 52.5 meters.
the largest area would then be 52.5^2 = 2756.25 square meters.
your old fence length would be 30 meters.
your new fence length would be 3 * 52.5 meters plus (52.5 - 30) = 22.5 meters.
3 * 52.5 + 22.5 = 180 meters of new fence connected to 30 meters of old fence.


without knowing the length of the old fence, the first two assumptions above would be the only logical assumptions i think i could make.


take your pick as to which of those assumptions you think the person who provided you with the probleman meant.


let me know if a difference assumption needed to be made.