Math CentralQuandaries & Queries


Question from Christy, a student:

Show that among all rectangles with an 8m perimeter, the one with the largest area is a square.

I know this is simple but I'm not sure if I'm doing it correctly. Here is what I did.

1. A = xy
2. 8 = 2x+2y
3. y = 4-x
4. A = x(4-x) = 4x-x^2

Not sure what to do from this point because I don't know if its right.



What you have done so far is correct. What you do next depends on what you know. Is this a calculus problem? If so the differentiate the function A(x) = 4x - x2 and solve A'(x) = 0 to find any critical points. This gives x = 2 and what remains is to show that x = 2 actually yields a maximum.

If this is not a calculus class then you might have an expression that tells you where the maximum or minimum of a parabola appears. You might notice that the parabola A = x(x - 4) crosses the x-axis at x = 0 and x = 4 and then conclude by the symmetry of the parabola that the maximum must appear half way between 0x = 0 and x = 4.

If this doesn't help then write back.


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