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Question Number 5338 by FilupSmith last updated on 09/May/16

Commented by FilupSmith last updated on 09/May/16

I have n lines inside the interior of a  semi circle with radius r. All lines are  of length a.    Show that the sum of all lines is less than  or equal to the circumference.    That is:  na≤πr

$$\mathrm{I}\:\mathrm{have}\:{n}\:\mathrm{lines}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{semi}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{radius}\:{r}.\:\mathrm{All}\:\mathrm{lines}\:\mathrm{are} \\ $$$$\mathrm{of}\:\mathrm{length}\:{a}. \\ $$$$ \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{lines}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than} \\ $$$$\mathrm{or}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{circumference}. \\ $$$$ \\ $$$$\mathrm{That}\:\mathrm{is}: \\ $$$${na}\leqslant\pi{r} \\ $$

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