| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
I'm having trouble with a series problem, maybe somone can help...
I have 39 boxes, stacked upon one another. Call the bottom one C1 and
the top C39.
The height of each box varies by some expansion factor with referece
to the one before it. (some sort of a log scale)
C1 = C1
C2 = C1 x (factor)
C3 = C2 x (factor)
= C1 x (factor)^2
C4 = C3 x (factor)
= C1 x (factor)^3
OK, now the stack has to be 0.833333 feet high, and
C39 must be about 3.32x10-3 feet high.
What should my expansion factor be?
Thanks,
Steve (who needs help on this one real quick)
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1293.1 | GUESS::DERAMO | Dan D'Eramo | Fri Sep 07 1990 12:14 | 44 | |
The height of the stack is
C1 + C2 + C3 + ... + C39
= C1 + C1 x + C1 x^2 + ... + C1 x^38
= C1 ( 1 + x + x^2 + ... + x^38 )
You can simplify the sum in parentheses by the formula
{ n (x = 1)
{
(1 + x + x^2 + ... + x^(n-1)) = {
{
{ (x^n - 1) / (x - 1) (not (x = 1))
Well, if x = 1, then the sum is 39 C1 = 39 C39 = 0.12948
which isn't 0.833333, so x must not be 1 (it will be between
0 and 1).
So you get
833333 = C1 * (x^39 - 1) / (x - 1)
C39 = C1 x^38 = 3.32x10-3
Now you could solve that as a 39th degree polynomial, but
I did an iterative solution in LISP and came up with a sum
of 0.8352 at x = 0.9242 and a sum of 0.8327 at x = 0.9243.
So the scale factor x would be between 0.9242 and 0.9243.
Running it a little longer I ended with x =
0.924276060875074112284362991508866L0, where at ten places
0.9242760608L0 ==> 0.833333001874303884480768725589584L0 and
0.9242760609L0 ==> 0.833332999377699094030607950158675L0,
although given the way I computed the sum:
(DEFUN F (X)
(SETQ C1 (/ C39 (EXPT X 38)))
(/ (* C1 (1- (EXPT X 39))) (1- X)))
I wouldn't trust all of that "precision".
Dan
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| 1293.2 | THANKS! | EUCLID::OWEN | Rent-to-own a clue | Fri Sep 07 1990 12:31 | 1 |