Question

Among all pairs of numbers (x, y) such that 2x + y = 20, find the pair for which the sum of the squares is a minimum.

Answer 1

I believe the sum of squares is a minimum when x=8 and y=4, if someone would check my work. Not 100% sure.

Answer 2

let the sum of the squares be...\[S = x ^{2}+y ^{2}\]

Answer 3

for all pairs of numbers (x,y), using original equation and solve for y...
\[y=20-2x\]

Answer 4

substitute y in our equation for S...\[S=x ^{2}+(20-2x)^{2}\]

Answer 5

simplifying S will lead to...\[S=x ^{2}-40x+200\]

Answer 6

getting the first derivative for minimum sum of squares...\[\frac{ dS }{dx }=2x-40\]

Answer 7

let dS/dx = 0 for minimum... we can solve for x...

0 = 2x - 40
2x = 40
x = 20

Answer 8

Oh I FOIL'd incorrectly. So could you plug in x = 20 into the original equation?

\[y = 20 - 2 \times 20\]

Answer 9

y = -20, so the pair of numbers is (20, -20) ?

Answer 10

yup...

Answer 11

thanks again! Just so you know, I am taking college algebra so instead of using the first derivative I plugged in the quadratic formula (minus the discriminant) into

\[- \frac{ b }{ 2 \times a}\]

which gives me the x value as well.

Answer 12

actually we were taught i remember using delta S/delta x before during my college days but not yet calculus... i forgot the steps involved..

Answer 13

ah okay, well it cant hurt for me to know the calculus way of doing it too. thanks again, I learned alot.

Answer 14

thanks also... maybe i'll be out for the meantime... nice chatting with you... :)

Answer 15

have a good one :)