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I have an optimization problem that I have not been able to solve on my own for the past week or so now. I need to find the largest rectangle given an x dimension that can fit inside of a circle with $r=12$, but also has to fit inside of $x > -6.307$ and $y > 3.125$. If this were just finding the maximum rectangle from the center this would be a lot easier, but since the "center" is technically at $(-6.307,3.125)$ I cannot figure it out.

Edit: There is also an upper bound of $y < 10.375$.

enter image description here

Photo for reference, rectangle has to fit inside the white area.

M. Wind
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Tanner Hernandez
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  • For some basic information about writing mathematics at this site see, e.g., here, here, here and here. – Another User Aug 11 '22 at 22:31
  • The largest rectangle in the white area will clearly use the lower left corner. The opposite corner will lie on the circle. Write the formula for the area as a function of the x coordinate of the opposite corner. Use calculus to find the maximum. Be sure to check the endpoints. I suspect that the maximum is at the intersection near $(12,10)$. – Ethan Bolker Aug 11 '22 at 22:58
  • Clearly the lower left corner of the rectangle is the lower left corner of the white area. The upper right corner is on the arc of the circle. Write an equation for the area of the rectangle as a function of the $x$ position of the upper right corner. Differentiate, set to zero,.. – Ross Millikan Aug 11 '22 at 22:58
  • This question says that it has to fit inside the white area, but that white area seems to have an upper bound of the form $y<10.4$ or so which is not mentioned in the question, as well as an upper bound for $x$ given by the lilac coloured region on the right. – Suzu Hirose Aug 11 '22 at 23:52
  • Surely there is an upper bound of $y<10.375$ and $x<10$? – Suzu Hirose Aug 12 '22 at 02:29

1 Answers1

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Take these coordinates for the four corners of the rectangle:

$A = (-6.307, 3.125)$ $B = (x, 3.125)$ $C = (x, y)$ $D = (-6.307, y)$

Assume that point $C$ is on the circle, so that $y$ in terms of $x$ is given by $y = \sqrt(144 -x^2)$. The surface area of the rectangle in terms of $x$ is given by:

$S = (x + 6.307)*(\sqrt(144-x^2) -3.125)$.

The maximum of $S$ can be found numerically by varying $x$. While doing so, check that $y < 10.375$ and that point $C$ is within the circle, i.e. $6.307^2 + y^2 < 144$. When $S$ is maximal and the two additional criteria are met, you have found the solution you sought.

I just did the calculation. Turns out the solution is symmetric around the $y$-axis. Optimal is $x = 6.307$, $y=10.209$, $S=89.356$.

M. Wind
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  • Your x and y values are very slightly outside the given radius of 12 units. Also, I got a maximum area value of 89.4317 at x=6.14 and y=10.31. – Suzu Hirose Aug 13 '22 at 04:18
  • My values were presented with a three decimal accuracy. Indeed $y = 10.2089$ would be better. But are you sure about your own result? If at point $D$ by Pythagoras the maximum value is $10.2089$ how can you justify the larger value $y = 10.31$ ? – M. Wind Aug 13 '22 at 13:43
  • Maybe it's wrong then, I didn't post an answer, I'll have another look. – Suzu Hirose Aug 13 '22 at 13:50