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A triangle has two corners, $(8,0,3)$ and $(0,8,3)$ and a third curve in space that consists of all points $(8,8,a^{2}+3)$, where $a$ is a real number. Calculate the area of the triangle as a function of $a$, $f(a)$ and determine its minimum value.

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Well, I have solved it. No need to worry. – user0233 Apr 18 '13 at 19:12
Then, you maybe want to answer your own question? – gerw Apr 18 '13 at 19:13

Find the length of each of the 3 sides: $\sqrt{8^2+8^2} = 4\sqrt{2}, \sqrt{8^2+a^2}, \sqrt{8^2+a^2}$ and use Heron's formula:

$$ s = 2\sqrt{2} + \sqrt{8^2+a^2} f(a) = \sqrt{s(s-4\sqrt{2})(s-\sqrt{8^2+a^2})^2} $$

anf you can minimize $f^2(a)$ for convenience.

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