SOLUTION: find two numbers whose sum is 30, such that the sum of the square of one number plus ten times the other number is a minimum. Thank you in advance for helping me with this funct

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Question 175552This question is from textbook college algebras
: find two numbers whose sum is 30, such that the sum of the square of one number plus ten times the other number is a minimum.
Thank you in advance for helping me with this function. This question is from textbook college algebras

Answer by eggsarecool(46) (Show Source):
You can put this solution on YOUR website!
The first part of this problem is interpretation.
two numbers whose sum is 30 means , since we don't know wither number we need two variables.
the square of one number plus ten times the other number means I am calling the equation f(x) so we have a name for it.
The next thing to remember is that to find the minimum of a function you take the derivative(calculus), set the derivative equal and solve.
Since they wanted a minimum for "such that the sum of the square of one number plus ten times the other number is a minimum" we need to take the derivative of f(x)
This presents a small problem since you can only have one variable when taking the derivative. We solve this by rewriting as
We then substitute this into f(x), giving us I replaced the with .
We now have
The next step is to take to derivative. , the derivative of was , the derivative of 300 since t is just a number is 0, and the derivative of is -10, giving you as your derivative
You then take and solve

and

Plugging back into you now have giving you
So the answer is x=5 and y=25 will give you the minimum