Math410 (Bayly) Homework 3 (due I forget when)

Section 2.4: 13, (19)

Section 2.5: (6), 7, 8, (9)

Section 3.4: 2, 3, (6), 13, (14), (16)

Section 4.2: (1), 3, (10), 15

Section 4.3: (5)

AND I looked up the PAC-10 scores online, and decided that for a 1-point homework problem it's too much work to ask you to analyze the actual PAC-10 rankings. So THAT will become an extra credit problem. Meanwhile for the homework I'd like you to rank teams A, B, and C which play against each other and have the following scores:

A 7 B 3,

A 4 C 8,

and B 6 C 4.

So if a, b, and c are some kind of "quality" of teams A, B, and C, then these scores mean a-b=4, c-a=4, and b-c=2. Clearly there is no solution, but you can seek a least-squares best approximation, and use those values of a, b, and c to rank the teams.

EXTRA CREDIT

I posted a handout on the theory of structures in equilibrium – pretty much the same as what I described in class that one day. For extra credit I would like you to

(1) Verify that the null vectors I gave for the 3-beam roof are always null vectors, whatever the locations of the beam ends. How would you generalize them for an arbitrary structure?

(2) For the 6-beam roof at the bottom of the handout, let's make it specific by assuming that the 4th corner is at (1,-1)^T. Also imagine you make it a 4-beam quadrilateral by removing the diagonal beams "c" and "e". Find the matrix for this configuration, and find a fourth null vector corresponding to a flexible deformation of the structure.

(3) Reinsert beam "c" and solve for the (hopefully unique) equilibrium, assuming the weight on the top is 600 pounds and a weight of 200 pounds on the bottom vertex (perhaps a hanging chair with a big guy in it).

(4) Reinsert beam "e" and solve for the equilibrium. I suspect you will find that tau_e will be a free variable, and I'd like you to express the other tensions in terms of tau_e. If tau_e=0, do you recover the solution in part (3)?

You may use your calculator or Rychlik's Java applet to do the matrix manipulations.

You can wait until after Spring break to turn this in.