You have 4 feet of wire to create a square and a circle. How much wire should you spend on each shape to minimize the area.

Also, why isn't the minimum area when you use all of the wire on the square, considering the square is less efficient than a circle in enclosing area? In other words, why is it better to spend all of the wire to make a square? Why wouldn't it always be better to make a circle to help improve efficiency. Another way to ask my question, why is the relationship between the two variables parabolic and not linear?

Thank you so much for anyone who can answer this. I've been stumped for a while.

  • $\begingroup$ If you are on the page where your question is, you will find, on the right, under Related, at least two questions that are essentially the same. I am sure there are several more on MSE, it is a standard max/min problem. It turns out that for max you give everything to the circle. That is also intuitively obvious. For minimum area, you use some square, some circle. This is because it is wasteful to surround two things. $\endgroup$ – André Nicolas Nov 15 '13 at 3:14
  • $\begingroup$ (Cont) Details for minimum, how much square and how much circle are I think not obvious. $\endgroup$ – André Nicolas Nov 15 '13 at 3:21
  • $\begingroup$ Why is it "wasteful to surround two things". Can you say that more mathematically? Can you tell me why it's a parabolic relationship and not a linear, where the less you spend on the square, the more area you get in total? $\endgroup$ – user108941 Nov 15 '13 at 3:26
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    $\begingroup$ Do a computation with say $32$ cm of wire. If I use it to make one square, area is $64$. If I make $2$ squares, each will be $4\times 4$, total area 432$. As to why the max is a circle, it is general result, isoperimetric inequality, circle is best. Look under Dido's problem. $\endgroup$ – André Nicolas Nov 15 '13 at 3:33
  • $\begingroup$ Thank you for your explanation. That makes sense. The part about the circle being more efficient was intuitive for me, I just couldn't wrap my head around why it is worse to waste wire on a square and a circle than to make one big square. I understand now. $\endgroup$ – user108941 Nov 15 '13 at 3:38

Say you use $x$ feet of wire for the square, then you are using $4-x$ feet of wire for the circle. Now you can express the area of both the square and the circle in $x$ (Hint: first compute the length of a side of the square and the radius of the circle). This gives you an expression in $x$ for the sum of the areas, which you have to minimize. Do you know how to do this?

  • $\begingroup$ Thank you for your response. Can you answer the second part of my question? $\endgroup$ – user108941 Nov 15 '13 at 3:21
  • $\begingroup$ The general point is that is it more efficient to enclose one large area than to enclose several smaller areas. For example, you need $8$ units of wire to enclose a $2 \times 2$ square (area $4$), but $16$ units of wire to enclose four $1 \times 1$ squares (with a total area of $4$). To see that the relationship is parabolic (and not linear), you should just calculate the formula for the area, which turns out to be a quadratic function. $\endgroup$ – Arthur Nov 15 '13 at 3:30
  • $\begingroup$ That makes sense. Thank you. $\endgroup$ – user108941 Nov 15 '13 at 3:38

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