All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
Star Stack (Posted on 2004-04-12) Difficulty: 3 of 5
Three pentagram-shaped stars (the stars formed from the diagonals of a regular pentagon) are stacked up so that the bottom two ends of the tips touch the middle ends of the tips of the star below. (See diagram.)

The distance from the top of the stack to the floor (where the bottom star's "feet" rest) is 4 feet.

What is the distance between the bottom two ends of the tips of the stack that touch the floor?

No Solution Yet Submitted by Gamer    
Rating: 3.5000 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
check | Comment 7 of 19 |

Thanks for the correction. I will check my answers more closely next time. With the correction, do I get the same answer as you?

We can represent the height of one of the triangles (i.e., points) of the big star in terms of s (a side, or long line of the big star) as

    ht(big) = s*sin72/(2+2cos72)

This is because s = perimeter of one of the "point-triangles".

Using ratios again,

ht(med) = ht(big)*s(med)/s
        = s*sin72*cos72/(1+cos72)
ht(lit) = ht(med)*s(lit)/s(med)
    = (s*sin72*cos72/(1+cos72)) * (4s*cos72*cos72)/2s*cos72
    = 2s*sin72*cos72*cos72/(1+cos72)

So now we use the correction,
h(lit) - ht(lit) + h(med) - ht(med) + h(big) = 4

4s*cos72*cos72*sin72 -
   2s*sin72*cos72*cos72/(1+cos72) +
      2s*cos72*sin72 -
         s*sin72*cos72/(1+cos72) +
            s*sin72 = 4

4s*cos72*cos72*sin72*(1+cos72) - 2s*sin72*cos72*cos72 +
 2s*cos72*sin72*(1+cos72) - s*sin72*cos72 +
   s*sin72*(1+cos72) = 4*(1+cos72)

s = (4*(1+cos72)/sin72) * (1/(4cos72*cos72*cos72 +  4cos72*cos72 + 2cos72 + 1))

We still want to find b(lit).

  b(lit) = 8s*cos72*cos72*cos72

substituting for s

b(lit) =  (32*cos72*cos72*cos72*(1+cos72)/sin72) *
       (1/(4cos72*cos72*cos72 + 4cos72*cos72 + 2cos72 + 1))

b(lit) = 0.613625084 feet




  Posted by vectorboy on 2004-05-24 14:51:52
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (2)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (9)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information