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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)

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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
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