More Puzzles

   Log inLog in 
 
 RegisterRegister Immediately 

Mathematical Puzzles
write 271 as the sum of positive real numbers

 
Sat Dec 04, 2010 6:10 pm  by tartle

write 271 as the sum of positive real numbers so as to maximize their product.
 
   
Mon Jan 03, 2011 1:13 pm  by s.b.

2+3+4+5+6+7+8....
note:0 is not applied cuz then product becomes 0. 1 does not change the product...am i right?
 
   
Sun May 29, 2011 7:48 pm  by bds021

135.5+135.5
 
   
Sat Jun 04, 2011 7:03 am  by Unni

3+2+2+2+2+2.......

Product = 3 x 134th power of 2
 
   
Sat Jun 04, 2011 7:05 am  by Unni

3+2+2+2+2+2.......

Product = 3 x 134th power of 2
 
   
Sun Jun 19, 2011 6:05 pm  by DiamondSoul

It's 2.71 repeated 100 times.
 
   
Wed Jul 06, 2011 4:28 pm  by cat

[quote="DiamondSoul"]It's 2.71 repeated 100 times.[/quote]

This appears to be correct, but needs a little explanation.

([i]n[/i]+1)*([i]n[/i]-1) = [i]n[/i]^2-1, which is less than [i]n[/i]^2. This shows that uniform values adding to a given sum make the largest product. Therefore, using [i]a[/i] for 271, the product [i]y[/i] of [i]x[/i] uniform values can be written as:

[i]y[/i] = ([i]a[/i]/[i]x[/i])^[i]x[/i] = e^[[i]x[/i]*(ln[i]a[/i]-ln[i]x[/i])]

The derivative is:

[i]dy[/i]/[i]dx[/i] = [([i]a[/i]/[i]x[/i])^[i]x[/i]]*[ln[i]a[/i]-(1+ln[i]x[/i])] = [([i]a[/i]/[i]x[/i])^[i]x[/i]]*[ln([i]a[/i]/[i]x[/i])-1]

At the maximum product, the derivative equals zero, so that:

ln([i]a[/i]/[i]x[/i]) = 1;
[i]a[/i]/[i]x[/i] = e;
[i]x[/i] = [i]a[/i]/e = 271/2.718... = 100 to the nearest integer.
 
   
Reply to topic
      All times are GMT
Page 1 of 1

 
 



Discussion Board Forum Index