## FoolForMath 2 years ago Another super easy problem, If an ant wants to crawl over the rectangular block of dimensions $$6\times5\times4$$ from one vertex to a diagonally opposite vertex, what is the shortest distance it would need to travel?

1. binary3i

$\sqrt{125}$

2. FoolForMath

No.

Is it a flying ant?

4. FoolForMath

No, no :)

5. apoorvk

4 + sqrt(61)

6. ninhi5

sqr 74

7. FoolForMath
8. Callisto

No......

9. FoolForMath

10. ninhi5

sqr 117

11. FoolForMath

Bingo ninhi5! $$\sqrt{117}$$ is the right answer.

12. ninhi5

hooray

13. Arnab09

15

14. apoorvk

The ant needs to travel one side, and one diagonal. We have three cases: side '4' + diagonal of (6 and 5) = 4 + sqrt61= 11.something side '6' + diagonal of (4 and 5) = 6 + sqrt41= 12.something side '5' + diagonal of (4 and 6) = 5 + sqrt52 =12.something hence shortest = 4 + sqrt61

15. apoorvk

Now where am I wrong?

16. SmoothMath

|dw:1338100316725:dw|$\sqrt{x^2 + 5^2} + \sqrt{(6-x)^2 + 4^2}$ By the pythagorean theorem, the sum of those two diagonals is: To minimize this distance, derive and set equal to 0.

17. ninhi5

dont make it over complicated :)

18. apoorvk

Oh damn! I completely forgot that !! :/ Damn my soul

19. FoolForMath

There are only 3 possible roots, consistent to Smooth's diagram.

20. binary3i

|dw:1338145571548:dw|

21. Ishaan94

11.7? lol

22. Ishaan94

Oh No :(

23. FoolForMath

@ninhi5: Can you post the explanation?

24. SmoothMath

I let Alpha do the derivation and optimization for me. http://www.wolframalpha.com/input/?i=solve+%28x%2F%28sqrt%28x%5E2%2B25%29%29+-+%286-x%29%2F%28sqrt%28%286-x%29%5E2%2B16%29%29%29+%3D0%2Cx

25. binary3i

$\sqrt{117}$

26. ninhi5

i just use pythogorean theorem

27. SmoothMath

So x = 10/3, giving this distance: http://www.wolframalpha.com/input/?i=sqrt%28%2810%2F3%29%5E2%2B5%5E2%29%2Bsqrt%28%286-%2810%2F3%29%29%5E2%2B4%5E2%29

28. ninhi5

My new question guys