Hi. I have a question that reads " Among all pairs of numbers whose sum is 15, find the pair such that the sum of their squares is the smallest possible."
Could someone please work this problem out for me and explain the steps. Thanks!!!!!
since there are not many numbers, let's try and do this by trial and error
here are the pairs of integers who's sum give 15
1 + 14 ...........1^2 + 14^2 = 197
2 + 13 ...........2^2 + 13^2 = 173
3 + 12 ...........3^2 + 12^2 = 153
4 + 11 ...........4^2 + 11^2 = 137
5 + 10 ...........5^2 + 10^2 = 125
6 + 9 ............6^2 + 9^2 = 117
7 + 8 ............7^2 + 8^2 = 115 ........this is your guy
and they repeat with the second number taking the place of the first and the first taking the place of the second
i'm sure TPH or someone else can come up with a neater way to do this
Call one of the numbers x, the other must then be 15 - x
As x + 15 - x = 15
Squaring them and adding them together we get
x^2 + (15 - x)^2 = x^2 + 225 - 30x + x^2 = 2x^2 - 30x + 225
Call this expression y, ie y = 2x^2 - 30x + 225
We are looking for the minimum point so differentiate
dy/dx = 4x - 30
For minimum dy/dx = 0 so 4x - 30 = 0
Solving
4x = 30
x = 7.5
So pair is 7.5, 15 - 7.5 = 7.5
let me try:
we want x + y = 15 and x^2 + y^2 a minimum
Calculus Way:
since x + y = 15
=> x = 15 - y
so x^2 + y^2 = (15 - y)^2 + y^2
= 225 - 30y +y^ + y^2
= 225 - 30y + 2y^2
Let's call this S
that is, S = 225 - 30y + 2y^2
since this is a parabola, we want S' = 0 for min point
=> S' = -30 + 4y = 0
=> 4y - 30 = 0
=> y = 30/4 = 7 1/2
but x = 15 - y = 15 - 7 1/2 = 7 1/2
so x = y = 7.5 is a solution, but these are not integers, so i guess we can shift 0.5 from 1 to the other to get the integers, so the numbers are 8 and 7. if you didn't necessarily want integers, then 7.5 and 7.5 is your answer
PreCalculus Way:
since x + y = 15
=> x = 15 - y
so x^2 + y^2 = (15 - y)^2 + y^2
= 225 - 30y +y^ + y^2
= 225 - 30y + 2y^2
Let's call this S
that is, S = 225 - 30y + 2y^2
we want the vertex of S
that is, we want, y = -b/2a = 30/2(2) = 30/4 = 7.5
but x = 15 - y = 15 - 7.5 = 7.5
and the same argument holds