SOLUTION: Find two numbers whose sum is 60, such that the sum of the square of one number plus twelve times the other number is a minimum.

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Question 329197: Find two numbers whose sum is 60, such that the sum of the square of one number plus twelve times the other number is a minimum.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
1.X%2BY=60
You are trying to minimize the value of
2.Z=X%5E2%2B12Y
From eq. 1,
Y=60-X
Substitute into eq. 2,
Z=X%5E2%2B12%2860-X%29
Z=X%5E2-12X%2B720
Complete the square to put the equation in vertex form,y=a%28x-h%29%5E2%2Bk
Since the coefficient of X%5E2 is positive, the vertex y value(k) will be the minimum.
Z=X%5E2-12X%2B36%2B720-36
Z=%28X-6%29%5E2%2B684
X=6
Y=60-6=54
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highlight%28X=6%29 and highlight%28Y=54%29 would provide the minimum value of 684.