HW5 - 60M - S06 - MW

Handed out F 5-19
Due        Th 5-25 in class
Hand in printout of active work sheet


1) A parameterization of the Umbilic Torus NC is given by 
t(s,t) = 
     x(s,t) [1,0,0]' + y(s,t) [0,1,0]' + z(s,t) [0,0,1]'
where -pi <= s,t <= pi,
and x = [7+cos(s/3-2t)+2cos(s/3+t)] sin(s)
    y = [7+cos(s/3-2t)+2cos(s/3+t)] cos(s)
    z = sin(s/3-2t)+2sin(s/3+t)
Graph the torus (3 dimensional parameterizes plot)

Produce some interesting variation of the torus.

2) The Beam Problem: Find the exact length of the longest
beam that can be carried arround a corner from a two feet
wide hall way to a hall way that is three feet wide.

3) Find a function eta(r) s.t.

 (eta(r) + r) / [2 ln (2/ (1+exp(-eta(r))) ]  
    \leq       1 + sqrt(r) + r/(2*ln(2))
  for all r>0

Hint: You might try eta(r) = sqrt(c r), for various choices of c.
Different c work for different ranges of r.

Prove with Maple that your choice of eta(r) does the trick.
Visualize your argument with various plots.

(This is tricky ... will give partial credit.)