Find the largest possible rectangular area you can enclose, assuming you have 128 meters of fencing. what is the (geometric) significance of the dimensions of this largest possible enclosure? My quesiton is what is the geometric significance and how do you find it?
I would use calculus to solve this question.
If you have a given rectangle with length x and width y, then $$2x + 2y = 128$$
And we are trying to maximize $$A(x) = xy$$ subject to that constraint.
If we solve for x in terms of y, we get that $$x = 64-y$$
Substituting this into our equation A(x) we get that $$A(x) = y(64-y)$$
In our to find the maximum we take the derivative and set it equal to 0: $$A'(x) = 64 - 2y = 0$$
Therefore, $$y = 32$$
And x = 32 as well.
In terms of geometric significance, if you want to maximize the area, always use a square.