Homework #9 solutions

HW #9, due Friday 4/14: 7.2 #8,14,19,20,25,27,31

7.2#8

Three hills, at about (3,7), (7,2), and (0,8). The third one is highest with height 8. The river runs downhill (to lower numbers) so northeast.

#14

f(x,y)=x+y has contours which are straight lines, with slope -1.

#19

f(x,y)=y-x^2 has contours which are all shifts of the same parabola, each left-right symmetric about x=0.

#20

  • a) I looks good for a highway, along a ridge.
  • b) II looks good for a canal, being a trough.
  • c) II or III both seem good - the track goes through a valley.

    • #25
  • a) The A path crosses lines closer together, so ascends more quickly.
  • b) The B line goes up a ridge, whereas A goes up a trough; so B will be able to see more.
  • c) Since A's in a trough, it's more likely to be along a stream.
    • #27
  • a) Only IV has happiness always increase with more pizza or cola.
  • b) Only I eventually decreases with either too much pizza or cola.
  • c) III eventually decreases with cola, but always increases with pizza.
    • #31
  • a) The contour line meeting (p=4,t=0) is 12. The contour line meeting (p=4,t=3) is something like 7.5. The 2 contour line meets p=4 at about t=4.5.
  • b) Fix a time (vertical line): then output goes up as pressure increases.
  • c) Fix pressure (horizontal line): then output goes down with time.
  • d) It takes a long time (at pressure=3) to cross contour lines, so output initially changes slowly. Then one starts crossing many contour lines in the same amount of time, i.e. output is dropping faster. So: the physician can initially worry about other things, but knows that at 2+ hours she must address cardiac output, that it can't be put off longer.