| Author |
Message |
![[Post New]](/templates/default/images/icon_minipost_new.gif) 11 Jul 2008 23:16:25 IST
|
|
|
Write 271 as the sum of positive real numbers so as to maximize their product
|
[url=http://www.signaturebar.com/][img]
|
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 11 Jul 2008 23:21:56 IST
|
|
|
271=136+135 136*135=18360. Couldn't find any other.
|
God does not care about our mathematical difficulties. He integrates empirically. ~~~Albert Einstein (1879-1955)~~~~
To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.~~~Pierre de Fermat (1601-1665)~~~
     |
this reply: 0 points
(with 0 
in 0 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 11 Jul 2008 23:35:54 IST
|
|
|
rudra ur ans i swrng as it says real no n not integers !!!!!!!!!
no the answer should be 135.5
as 135.5*135.5 = 18360.25 and that is the max product
rate if satisfied!
|
It is not easy to see how the more extreme forms of nationalism can long survive when men have seen the Earth in its true perspective as a single small globe against the stars. Arthur C. Clarke
http://removesuperstition.blogspot.com/
|
this reply: 4 points
(with 0 
in 2 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 12 Jul 2008 00:19:53 IST
|
|
|
hey
u c itz positive real nos......
so d first thing u shud look at is
x + y/2 >= rt(xy)
so itz obviously (135.5)^2........
in maximisin probs the value u luk fer mostly lies in middle
|
dbznfreak---watchin episodes for 6 yrs--movin on to dbgt
<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0">
<TR><TD>
<DIV ALIGN="right">Animated Letters</DIV></TD></TR></TABLE>
|
this reply: 0 points
(with 0 
in 0 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 16:49:45 IST
|
|
|
oops........stupid me!
it s not mentioned 2 + real nos...is it........
let me assume the number of numbers to b 'n'
so
x1 + x2 + ....+ xn >= n (x1x2.....xn)1/n
.......pdt....<= (271/n)n
so pdt is a function of number of terms
so p(n) = (271/n)n.........
logp = n log(271) - nlogn
1/p * dp/dn = log271 - logn - 1
dp/dn = p[log (271 / n) - 1]
dp/dn = 0 ------->> log(271/n) = 1 or p=0
n = 271/e or p = 0
but p cant be zero so
n = 271/e
n x1 = nx2 = .......= nxn = 271
so...
x1 = x2 = x3=.......=xn = e
so the positive real numbers required are:
271/e................n times..........
x1 = x2 = x3=.......=xn = e.........
uhhhhhh........now anothr prob.....,m gettin not gettin n as an integer.......ne1 help.....
|
dbznfreak---watchin episodes for 6 yrs--movin on to dbgt
<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0">
<TR><TD>
<DIV ALIGN="right">Animated Letters</DIV></TD></TR></TABLE>
|
this reply: 10 points
(with 2 
in 2 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 17:10:16 IST
|
|
|
Let the numbers be x and 271-x 
P(x) = Product of two numbers
P(x) = x ( 271-x )
P'(x) = 271 - 2x
Since P(x) is max hence P'(x) = 0 
x = 135.5 
|
"Before you start some work, always ask yourself three questions - Why am I doing it, What the results might be and Will I be successful. Only when you think deeply and find satisfactory answers to these questions, go ahead."
Chanakya quotes (Indian politician, strategist and writer, 350 BC-275 BC) |
this reply: 0 points
(with 0 
in 0 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 17:15:56 IST
|
|
|
@sachin gupta.....
dats d mistake we all made d last tym.........
its not mentioned 2 nos.--got it!
accordin to ma soln.....the max pdt is e271/e ~~~approx....(2.7)100.....which is way higher than ur 135.5^2
xperts......reply
|
dbznfreak---watchin episodes for 6 yrs--movin on to dbgt
<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0">
<TR><TD>
<DIV ALIGN="right">Animated Letters</DIV></TD></TR></TABLE>
|
this reply: 7 points
(with 1 
in 2 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 17:29:44 IST
|
|
|
@vasanth i think you are correct. It is no where mentioned that it has to be 2 numbers. It can be more that that.
|
God does not care about our mathematical difficulties. He integrates empirically. ~~~Albert Einstein (1879-1955)~~~~
To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.~~~Pierre de Fermat (1601-1665)~~~
     |
this reply: 0 points
(with 0 
in 0 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 19:12:55 IST
|
|
|
Call the product for n integers as f(n)
Then 
You will observe that the function initially increases. We must find out, up to what n this happens.

Now, we can use tha fact that the function approaches e rapidly. So, we can approximate it by 2.7
This gives n = 100 as a good estimate for the largest such n. Thereafter it is a decreasing function.
I checked with Excel (that's the best I have!) and 100 is right
|
Time wounds all heels |
this reply: 10 points
(with 2 
in 2 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 19:23:22 IST
|
|
|
let the numbers b x and 271-x
their product is equal to 271x-x^2 now f(x)=271x-x^2
for the function to be maximum or minimum d/dx{f(x)=0
i.e 271-2x=0 =>x=271/2=135.5
now finding the double derivative of f(x)
it is equal to -2<0 this proves that x=135.5 is a point of maximum....
thus the two numbers are 135.5 and 135.5
further for a competitive point of view all the questions of this kind the answer is always half the number given.....
rate if useful....
|
this reply: 0 points
(with 0 
in 0 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 19:25:31 IST
|
|
|
ans all are 2.71
Proof : Using the old A.M>= G.M
G.M. <= ( 271/n) ^n ( it attains the maximum value when all xi's are equal )
Now consider the characteristics of ( 271/x)^x ( let x be a real no )
Now we have that it attains its maximum value at x= 271/e= 99.6959
we need to consider only n = 99 or, n =100
Trough actual calculation it comes out that for x= 100 it attains the maximum value
Hence all the lenghths should be 2.71
|
this reply: 5 points
(with 1 
in 1 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
![[Post New]](/templates/default/images/icon_minipost_new.gif) 13 Jul 2008 19:26:22 IST
|
|
|
@mukulaish: The question has not placed a restriction on the number of real numbers that can be there.
|
Time wounds all heels |
this reply: 0 points
(with 0 
in 0 votes ) [?]
|
|
You have to be logged on to rate
|
|
|
|
|