Chaya Galloway

2021-08-14

Find two positive numbers whose sum is 676 and the sum of whose squares is a minimum.

SoosteethicU

Skilled2021-08-15Added 102 answers

Let the two numbers be x and y.
Sum is 676
x+y=676
Solve for y
y=676-x
Now, sum of squares
$$={x}^{2}+{y}^{2}\phantom{\rule{0ex}{0ex}}={x}^{2}+(676-x{)}^{2}\phantom{\rule{0ex}{0ex}}={x}^{2}+456976+{x}^{2}-1352x\phantom{\rule{0ex}{0ex}}2{x}^{2}-1352x+456976\phantom{\rule{0ex}{0ex}}\text{where}\text{}a=2{x}^{2},b=1352x,c=456976$$
Minimum occurs at vertex
$$\text{So,}\text{}x=\frac{-b}{2a}\phantom{\rule{0ex}{0ex}}x=\frac{1352}{2(2)}\phantom{\rule{0ex}{0ex}}x=\frac{1352}{4}\phantom{\rule{0ex}{0ex}}x=338\phantom{\rule{0ex}{0ex}}y=676-x\phantom{\rule{0ex}{0ex}}y=676-338\phantom{\rule{0ex}{0ex}}y=338$$
So, (338, 338) is Answer

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