SOLUTION: Find two numbers such that their product is as large as possible, given that two times the first number plus three times the second number is 60

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Question 954660: Find two numbers such that their product is as large as possible, given that two times the first number plus
three times the second number is 60
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two numbers be X and Y.
You want to maximize Z=X%2AY with 2X%2B3Y=60.
From the second equation,
2X=60-3Y
X=30-%283%2F2%29Y
Substitute into the product,
Z=%2830-%283%2F2%29Y%29Y
Now the product is the function of only one variable.
You can either take the derivative and find the extrema or since it's a quadratic, you can convert it to vertex form to find the maximum.
Z=-%283%2F2%29Y%5E2%2B30Y
Z=-%283%2F2%29%28Y%5E2-20Y%29
Z=-%283%2F2%29%28Y%5E2-20Y%2B100%29%2B%283%2F2%29%28100%29
Z=-%283%2F2%29%28Y-10%29%5E2%2B150
In vertex form, the parabola opens downwards and the vertex value is the maximum.
The maximum Z=150 occurs when Y=10
So then,
X=30-%283%2F2%29%2810%29
X=30-15
X=15