Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.

The point on the graph shows his
arrival at the bumper cars. What do
the coordinates of the point tell us
about the situation?

What does the point (0,0) mean in
this situation?

What is the constant of
proportionality for the relationship
between time and distance? What
does it tell you about Tyler's walk?
Where do you see it in the graph?

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  a) It took 40 seconds to go 50 meters from the ticket booth

  b) Tyler started at the ticket booth

  c) k = 5/4

  d) Tyler walked at the rate of 1.25 meters per second

  e) The constant of proportionality is the slope of the line from (0, 0) to (40, 50). It is not shown on the graph.

Step-by-step explanation:

a) The axis labels on the graph tell you that 40 represents the number of seconds of elapsed time, and 50 represents the number of meters from the ticket booth. With the information in the problem statement, the coordinates (40, 50) tell us it took Tyler 40 seconds to walk 50 meters from the ticket booth to the bumper cars.

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b) The point (0, 0) tells us that Tyler started counting time when he left the ticket booth.

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c) The constant of proportionality is the ratio of the second coordinate to the first. The constant of proportionality is 50/40 = 5/4 = 1.25. (Its units are meters per second.)

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d) The constant of proportionality tells us that Tyler's steady pace was 1.25 meters per second.

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e) The constant of proportionality is not shown on the graph. It is the slope of a line between the origin and the point (40, 50).

i think it's eleven too, don't woosh me here, but according to the "experts" it's 2.

the proof starts from the peano postulates, which define the natural

numbers n. n is the smallest set satisfying these postulates:

  p1.   1 is in n.

  p2.   if x is in n, then its "successor" x' is in n.

  p3.   there is no x such that x' = 1.

  p4.   if x isn't 1, then there is a y in n such that y' = x.

  p5.   if s is a subset of n, 1 is in s, and the implication

      (x in s => x' in s) holds, then s = n.

then you have to define addition recursively:

  def: let a and b be in n. if b = 1, then define a + b = a'

      (using p1 and p2). if b isn't 1, then let c' = b, with c in n

      (using p4), and define a + b = (a + c)'.

then you have to define 2:

  def:   2 = 1'

2 is in n by p1, p2, and the definition of 2.

theorem:   1 + 1 = 2

proof: use the first part of the definition of + with a = b = 1.

      then 1 + 1 = 1' = 2   q.e.d.

note: there is an alternate formulation of the peano postulates which

replaces 1 with 0 in p1, p3, p4, and p5. then you have to change the

definition of addition to this:

  def: let a and b be in n. if b = 0, then define a + b = a.

      if b isn't 0, then let c' = b, with c in n, and define

      a + b = (a + c)'.

you also have to define 1 = 0', and 2 = 1'. then the proof of the

theorem above is a little different:

proof: use the second part of the definition of + first:

      1 + 1 = (1 + 0)'

      now use the first part of the definition of + on the sum in

      parentheses:   1 + 1 = (1)' = 1' = 2   q.e.d.