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Two numbers have a sum of 13.  Find the minimum of the sum of their squares.

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alquigg | Student, Grade 11 | eNotes Newbie

Posted February 21, 2009 at 1:55 PM via web

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Two numbers have a sum of 13.  Find the minimum of the sum of their squares.

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bizzzye | Middle School Teacher | eNotes Newbie

Posted March 10, 2009 at 1:39 AM (Answer #1)

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Hi!

9+4=13

The square root of 9 is 3

The square root of 4 is 2

3+2=5

I hope that helps.

Thanks, bizzzye

 

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 13, 2011 at 2:45 AM (Answer #2)

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Let x and y be the numbers.

Adding the numbers yields 13: x+y = 13

The sum of the squares is `x^2 + y^2 = S`

Write the sum of squares in terms of x.

`S(x) = x^2 + (13-x)^2`

Expanding the binomial yields:

`S(x) = x^2 + 169 - 26x + x^2 =gt`  `S(x) = 2x^2 - 26x + 169`

The sum of squares is minimum if `S'(x) = 0` .

Calculating `S'(x)`  yields:`S'(x) = 4x - 26` .

If `S'(x) = 0 =gt 4x - 26 = 0 =gt 4x = 26 =gt x = 26/4`

`y = 13 - 26/4 =gt y = (52-26)/4 =gt y = 26/4`

If `x=y=26/4` , the sum of the squares of the numbers is minimum.

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