An analogy: a child with blocks and constraints on physical systems. a certain child's room is partitioned into 100 squares. his toy box in the corner is exactly one square in size and contains some number of blocks. when he plays with the toys, he tends to throw them around and evenly scatter them about the room.
a) assume that there are now three blocks (red, green, and blue). also assume that all three blocks can fit on one square. how many accessible states are there for this system, if they are all in the toy box, lid is closed, and the lid is locked in the closed position? (all the other constraints imposed in the story still apply.)
b) what is the total number of accessible microstates for this system, if the toy -box lid is opened?
c) if the boy plays for a long time with all the blocks (and randomly leaves them in the toy box or on one of the floor squares), what is the probability of finding the red block in the toy box, the blue box on square #15, and the green block on square #75? explain how you determined this result.
d) again, after a long time, what is the probability of finding the red block in the toy box independently of where the blue and green blocks are?
e) how would probability in (d) change if we asked for it after the boy had been playing for only 15 seconds? explain why.
f) explain what condition must be satisfied to say that a system is equally likely to be in any of its accessible microstates.
k = ⅔
d. y = ⅔xz; y(6, 3) = 12
y ∝ x and y ∝ z, so
y ∝ xz and
(1) y = kxz divide each side by xz
k = y/(xz) let x = 1, z =4, y = ⁸/₃
= (⁸/₃)/(1 × 4)
= ⁸/₃ × ¼ cancel 4s
(2) k = ⅔ substitute into (1)
y = ⅔xz let x = 6, z = 3
y = ⅔ × 6 × 3 cancel 3s
y = 2 × 6
y = 12
y(6, 3) = 12
x^2 - 8x + 30 = 3x
subtract 3x from each side
x^2 - 8x-3x + 30 = 3x-3x
x^2 - 11x + 30 = 0
what 2 numbers multiply to 30 and add to -11
-5*-6 = 30
-5+-6 = -11
(x-5) (x-6) =0
using the zero product property
x-5 = 0 and x-6 =0
x-5+5 =0+5 and x-6+6 =0+6
calculate the number of years 2009-2001 = 8 years
8 years x $35 each year = $280
$1950 - $280 = $1670