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anonymous

  • 5 years ago

what is the maximum area of a square that can be fit in a circle of radius 'r'?

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  1. anonymous
    • 5 years ago
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    Well what would the length of the diagonal line through the square be?

  2. anonymous
    • 5 years ago
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    if you assume the side of the square to be 'a' then the length of the diagonal will be \[\sqrt{2}a\] and assume the rquation of the circle be \[x ^{2}+y ^{2}=r ^{2}\]

  3. anonymous
    • 5 years ago
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    *equation

  4. anonymous
    • 5 years ago
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    Yes, but if the square is inscribed in the circle, the points of the corners will lie on the circle. Therefore the diagonal to the square will be the diameter of the circle or twice the radius.

  5. anonymous
    • 5 years ago
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    So we have that \[a\sqrt{2} = r \implies a = \frac{r}{\sqrt{2}}\] And you said that a was the length of the side, so using this we can find the area in terms of r. \(Area = a^2=\ ?\)

  6. anonymous
    • 5 years ago
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    you seem to have made a minor mistake.. shouldn't it be, \[a \sqrt{2}=2r\] and so the area is inevitably equal to \[a ^{2} = 2r^{2}\]

  7. anonymous
    • 5 years ago
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    If the square has a side a then 2a^2 = r or a = sqrt(r/2) max area = sqrt(r/2)*sqrt(r/2) = r/2

  8. anonymous
    • 5 years ago
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    a^2 + a^2 = r^2 a = sqrt(r^2/2) Area = a*a = r^2/2

  9. anonymous
    • 5 years ago
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    one more time a^2 + a^2 = 4r^2 a = sqrt(r^2/2) Area = a*a = 4r^2/2

  10. anonymous
    • 5 years ago
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    last time a^2 + a^2 = 4r^2 a = sqrt(4r^2/2) Area = a*a = 4r^2/2

  11. anonymous
    • 5 years ago
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    or Area = 2r^2

  12. anonymous
    • 5 years ago
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    Yes sorry, that's correct.

  13. anonymous
    • 5 years ago
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    good work ivan.. but using polpak's idea to find it out using the diagonal of the square is a hell lot a shorter

  14. anonymous
    • 5 years ago
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    \[a\sqrt{2} = 2r \implies a = \frac{2r}{\sqrt{2}} \implies a^2 = \frac{4r^2}{2} = 2r^2\]

  15. anonymous
    • 5 years ago
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    This also can be interesting. The area of a square with a side x is \[A(x) = x^{2}\] The distance of any two points inside of a circle must be less than or equal to 2r The longest distance in a square is the diagonal \[diagonal = \sqrt{x^{2} + x ^{2}} = \sqrt{2x ^{2}}\] so, we have: \[\sqrt{2x ^{2}}\le2r\] or \[x \le \pm \sqrt{2r ^{2}}\] \[A(x) = x ^{2}, D:\left\{ x | x \le \pm \sqrt{2r ^{2}} \right\}\] \[A \prime(x) = 2x, D=\left\{ x | x <\pm\sqrt{2r ^{2}} \right\}\] \[A \prime \prime(x) = 2, D=\left\{ x | x <\pm\sqrt{2r ^{2}} \right\}\] \[A \prime(x) = 0, for x = 0\] \[A \prime(x) > 0 , for x > 0\] \[A \prime(x) < 0 , for x < 0\] \[A \prime \prime(x) > 0, for all x\] => A(x) is concave up for all x \[=> A(x) is decreasing for-\sqrt{2r ^{2}} < x <0\] \[=> A(x) is increasing for: 0 < x < \sqrt{2r ^{2}}\] => A(x) has a minimum at x = 0, A(0) = 0 => A(x) desn't have a maximum value in the open interval \[(-\sqrt{2r ^{2}},\sqrt{2r ^{2}})\] => the maximum value of A(x) on the closed interval \[[-\sqrt{2r ^{2}}, \sqrt{2r ^{2}}] \]must be at\[ x \in \left\{ -\sqrt{2r ^{2}},\sqrt{2r ^{2}} \right\}\] => the maximum area of a square inside a circle is: \[A(\pm\sqrt{2r ^{2}}) =(\pm\sqrt{2r ^{2}})^{2} = 2r ^{2}\]

  16. anonymous
    • 5 years ago
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    boy! was that scary! lol

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