agentx5 3 years ago Mental Puzzle! ^_^ First, please view this question: http://openstudy.com/study#/updates/500c014fe4b0549a892fe7dc Now my question is this: How many combinations of WHOLE # units are there for the dimensions that can yield that total area? To be clear: No fractions. No roots. No irrationals. No imaginary #'s. Just strictly whole # units, how many combinations are possible? Have at it math people! I look forward to seeing your methods! (please don't just post your combination total, show us what you did to do it)

1. agentx5

For reference: an isosceles trapezoid with an area of 21540 ft$$^2$$ http://mathworld.wolfram.com/IsoscelesTrapezoid.html

2. agentx5

Make sense? ^_^

3. agentx5

Related example done, to show visually what I mean |dw:1342968040706:dw| 1m * 16m = 16m$$^2$$ is also valid, however... 2$$\sqrt{2}$$m * 4$$\sqrt{2}$$m = 16 m$$^2$$ gives the right area, sure; but is INVALID for this question, whole units only. $$\sqrt{2}$$ isn't a whole number :-)

4. agentx5

24 hrs later and nobody has attempted this? :-/

5. mukushla

hi agentx5 u mean to find all integers (a,b,h) such that $$(a+b) h=2*21540= 2^3 * 3 * 5 * 359$$

6. agentx5

Err I believe so yes... All positive integers that yield a final area from that formula that is 21540 units$$^2$$ :-)

7. agentx5

I'm just going to close it in a few if nobody is interested in trying the puzzle, it was more for fun & learning than "an answer for a homework problem" or anything like that. Saw the previous question (see link) and though it would be an interestingly related question. ;-)

8. mukushla

it seems we have a nice combination problem workin on it...

9. mukushla

oh oh but this is really a Mental Puzzle ; so many answers ; just with letting a+b=359 u have 358 answers...o.O

10. agentx5

358? Why one less?

11. agentx5

This type of geometrical combination puzzle is the kind the like what security systems use on the new-style touch-screen lockpads for vaults and such. The human user has to remember only a 3D shape, but an algorithm to solve it would be much harder. :-)

12. mukushla

nice... and 358 because a,b>0 a=1,2,...,358 consedering symmetry for a and b u have 358 answers

13. agentx5

14. agentx5

The height of the trapezoid?

15. mukushla

if u let a+b=359 then there is one valur for height; and that is 3*5*8

16. mukushla

value***

17. mukushla

number of divisors of area=8*3*5*359 is (3+1)(1+1)(1+1)(1+1)=32 h can be all of them like h=1 or 2 or 359

18. mukushla

after chossing 'h' u must find answers for a+b=(area/h) so there are many many answers its better to do it with computer i think ...

19. agentx5

Alright well I'm impressed at you for giving it the ol' college try :-D ty!

20. mukushla

yw :)