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The sum of a number and three times another number is 30. Their product is a maximum.
Determine the two numbers
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Let "another" number be x.
Then "the number" is 30-3x.
The product of the two numbers is x*(30-3x) = 3x*(10-x).
It is a quadratic function, represented by a parabola opened downward. Its roots (the x-intercepts) are 0 and 10.
The maximum is reached at the mid-point between the roots.
This mid-point is x= 5.
Thus "another" number is 5 and "the number" is 30-3*5 = 15.
ANSWER. The number is 15. Another number is 5.
Solved, with complete explanations.
