The numbers 2, 3, 5, 7, and 13 are placed on a number line. For any integer x, its "remoteness" is the sum of the squares of its distance to each of the five numbers. For example, the remoteness of 10 is 8² + 7² + 5² + 3² + 3² = 156. What integer has the smallest remoteness?
2006-12-03 12:29:13 · 3 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Just test numbers and find the smallest value:
r(x=4) = 2^2+1^2+1^2+3^2+9^2 = 96
r(x=5) = 3^2+2^2+0^2+2^2+8^2 = 81
r(x=6) = 4^2+3^2+1^2+1^2+7^2 = 76
r(x=7) = 5^2+4^2+2^2+0^2+6^2 = 81
r(x=8) = 6^2+5^2+3^2+1^2+5^2 = 90
Looking at this pattern, you can deduce that remoteness increases at x<6 and x>6. Thus, minimal remoteness is at x=6
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Hope this helps
2006-12-07 06:55:06 · answer #1 · answered by JSAM 5 · 0⤊ 0⤋
Let x be the number
need to minimize
(x-2)^2 + (x-3)^2 + (x-5)^2 + etc
to minimize, differentiate:
x-2 + x-3 + ... = 0
5x= 2+3+5+7+13 = 30,
x = 6
ta da!
2006-12-07 17:01:10 · answer #2 · answered by modulo_function 7 · 0⤊ 0⤋
Without taking a few minutes to just do the math, I would guess 8
2006-12-03 20:39:00 · answer #3 · answered by crazydave 7 · 0⤊ 0⤋