agentx5 Group Title 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) 2 years ago 2 years ago

1. agentx5 Group Title

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

2. agentx5 Group Title

Make sense? ^_^

3. agentx5 Group Title

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 Group Title

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

5. mukushla Group Title

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 Group Title

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

7. agentx5 Group Title

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 Group Title

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

9. mukushla Group Title

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 Group Title

358? Why one less?

11. agentx5 Group Title

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 Group Title

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

13. agentx5 Group Title

14. agentx5 Group Title

The height of the trapezoid?

15. mukushla Group Title

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

16. mukushla Group Title

value***

17. mukushla Group Title

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 Group Title

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 Group Title

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

20. mukushla Group Title

yw :)