Pretest
Answers
1.
(a) 1.5 x 10 5
(b) 3.6 x 10 12
(c) 2.5 x 10 8
(d) 10 -4
(e) 1.37 x 10 -1
2. (a) 
(b) Reasoning
as above gives
3. 
4. 
5. 
6. 
As in question 2, take ratios to get rid of all constants
note when all linear
dimensions are scaled by some factor, the volume increases or decreases by the
cube of that factor! (Surfaces areas
scale by the square of the factor!)
7. (a) 
(b) 0.25 rad
(c) 
8. (a) 
(b) 
(c) 
9. 
10. 
Using the
binomial theorem
11. Make a table
|
# weeks i |
# cockroaches N |
Double previous # |
Algorithm |
|
0 |
2 |
2 1 |
N=2 0+1 |
|
1 |
4 |
2(2 1) |
N=2 1+1 |
|
2 |
8 |
2(2 2) |
N=2 2+1 |
|
3 |
16 |
2(2 3) |
N=2 3+1 |
|
|
|
|
|
|
|
|
|
|
|
j |
6 x 10 27 |
2(2 j) |
N=2 j+1 |
Solve
Take
natural logs both sides
If I had chosen a lifetime for cockroaches that was less than 91 weeks, you would have to take this into account, obviously a much more complicated problem.
12. The equation for temperature as a function can be written as
where k = 0.5207
(a)
.. at midnite t = 0 
at 2 AM t
= 2 
(b) 
(Note the sign of the
exponent!)
So, the time of death was 10:52:16 and Clare was still making a nuisance of herself in the bar!
13. Construct another
square as shown with side b. Note 2
triangles of equal areas make up square a, while 4 make up square b. Thus, 
and 
14. (a) Length lo of the outside, circumscribed polygon
where 
. Here, we are
representing q
in radians
remember 
Perimeter Co of the outside polygon
Therefore, 
Likewise, the perimeter Ci of the inside, inscribed polygon
Therefore, 
Thus, we
obtain
(b) For n = 96
(c) 
You might note from above that
In other words, for small q 
15. Units of the period T of the pendulum must equal units of m a l b g c
Note that
units of m a are
and so on for the other variables, so
This means
that 
in order to balance
out the units.
Solving, we
get
This is as far as
dimensional analysis can take us. It
gives us the right variable dependency apart from dimensionless numerical
factors. The exact answer from fundamental principles of physics is
.
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