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In a rhombus of side length $$5$$, the length of one of the diagonals is at least $$6$$, and the length of the other diagonal is at most $$6$$. What is the maximum value of the sum of the diagonals?
If the diagonals are $$p$$ and $$q$$, then the half-diagonals and a side form a right triangle, so $$p^2+q^2=4\cdot5^2=100$$. Since one diagonal is at most $$6$$ and the other is at least $$6$$, the sum is maximized when the smaller diagonal is $$6$$. The other diagonal is then $$\sqrt{100-36}=8$$, giving a maximum sum of $$14$$.
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