SOLUTION: Two numbers are related by the equation 3x+ 2y = 42. Find the values of the numbers so that their product is a minmum

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Question 964813: Two numbers are related by the equation 3x+ 2y = 42. Find the values of the numbers so that their product is a minmum
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Find the minimum value of Z where Z is defined as,
Z=xy
.
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From the equation,
2y=42-3x
y=21-%283%2F2%29x
Substituting,
Z=x%2821-%283%2F2%29x%29
Z=21x-%283%2F2%29x%5E2
Convert Z to vertex form, then you can find the extremum.
Z=-%283%2F2%29x%5E2%2B21x
Z=-%283%2F2%29%28x%5E2-14x%29
Z=-%283%2F2%29%28x%5E2-14x%2B49%29%2B%283%2F2%29%2849%29
Z=-%283%2F2%29%28x-7%29%5E2%2B147%2F2
The value at the vertex is a maximum because of the -3%2F2 multiplier.
It occurs when x=7 and the maximum value is 147%2F2.
There is no minimum for the product, only a maximum.
Check your problem setup and repost if needed.
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