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“With a hundred inches of ink I draw a square and a circle so that the sum of the areas is a minimum. What is the length of the perimeter of the square? (Hint: Find the area as a function of p. Expect about a dozen lines of computation.)”

I know the answer is 400/(π+4), but I have no idea how to get there. I’ve been working for two hours but I don’t seem to be getting any closer.

brier
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  • Suppose the amount of ink used for the circle is $x$. Then the amount used for the square is $100-x$. Since $x$ is the circumference of the circle, its area must be $\frac1{4\pi} x^2$. Since $100-x$ is the perimeter of the square, its area must be $\frac1{16}(100-x)^2$. Do you have this? – Andrew Chin Jan 27 '20 at 20:03
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    Consider writing the equations for perimeter and area with respect to the radius of the circle and the length of the side of a square (i.e. two formulas with respect to $x$ and $r$). If you can do that, you'll be on your way. – scoopfaze Jan 27 '20 at 20:03
  • Where does the 1/16 come from? Maybe I’m just dumb, but I don’t understand how to rewrite the equations. ‍♂️ – brier Jan 27 '20 at 20:30
  • Does this answer your question? Maximize and Minimize a 12" piece of wire into a square and circle –  Jan 27 '20 at 21:25
  • I understand the 1/16 now, but I’m still not getting 400/(π+4) – brier Jan 27 '20 at 21:55

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Let $s$ be the length of the side of the square, and $r$ the radius of the circle. You know that $$ 4s+2\pi r=100. $$ And you want to minimize $$ s^2+\pi r^2=s^2+\pi\left( \frac{100-4s}{2\pi}\right)^2=s^2+\frac{(50-2s)^2}{\pi}. $$ If you differentiate with respect to $s$ and equate to zero, you get (after dividing by $2$) $$ s-\frac{2(50-2s)}\pi=0. $$ Solving for $s$, we get $$ s=\frac{100}{\pi+4}. $$ So that's the side of the square. To get the perimeter of the square, multiply by $4$.

We can confirm that this is a minimum either by checking the second derivative, or by noticing that we were minimizing a parabola, where there is a single critical point.

Martin Argerami
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  • Why does 4s + 2r = 400? 4s = the square’s perimeter. 2r = the circle’s perimeter. According to the question, the perimeter of the square plus the perimeter of the circle equals 100 inches of ink. Am I missing something? – brier Jan 28 '20 at 00:09
  • I misresd. I'll edit the answer. – Martin Argerami Jan 28 '20 at 00:50
  • And then, of course, there's the classic follow-up: what's the perimeter of the square which maximizes the sum of the areas ... . – Ned Jan 28 '20 at 02:43
  • Thank you so much! – brier Jan 28 '20 at 17:03