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I have a circle with radius $r$ and a square with side $a$. Their combined circumference is 20 meters. From this I need to decide the minimum and maximum area of the circle and the square together.

What I did was I wrote down the circumference $C = 4a+2\pi r = 20$ and the area $A = \pi r^2 + a^2$, am I right in thinking I can solve for either $a$ or $r$ from the circumference equation and then plug that in to the equation for A, to then find its minimum point?

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  • $\begingroup$ Should the radius and side of the square both be "a" in the first line? $\endgroup$ – Tyberius Apr 6 '17 at 20:25
  • $\begingroup$ Fixed it, thanks! $\endgroup$ – J.st Apr 6 '17 at 20:26
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Am I right in thinking I can solve for either $a$ or $r$ from the circumference equation and then plug that in to the equation for $A$, to then find its minimum point?

Yes. If you solve for $a$, for example, you get $A=(5-\frac{\pi}{2}r)^2+\pi r^2$. You can easily find the minimum and maximum on any given interval. You just need to identify the interval $r$ can belong to. It cannot be negative and the circle cannot take more than the whole circumference, so…

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