Optimization in problem solving and studying

[SpicyRamen]

This might be a stupid question, I'm not use to asking questions about math...

I just started on optimization. Can someone tell me What optimization is used for and how I could apply it to a problem. When it comes to a problem, I could do it like if it asks " Find two numbers that satisfy the given requirements,' The sum of (S) and the product is maximum.'" Like when They ask me that I can definitely do that no problem, but my teacher usually gets us with blinding questions that I have never seen before in our books, I know he's only trying to prepare us for the Ap exam, but sometimes No matter how much I study from the book I just don't know how to prepare for the test. He teaches us the material and I'm all like "Hey I totally get how to do this" But during the test he gives us problems we have never seen before and wants us to apply what we learn, so I want to prepare for the worst. I want to know why optimization is used. I know it's used to find max volume, minimum distance and area and min length, But why? Can someone give me a definition of optimization or show me using an equation how it originates.

 

[Feldoh]

There's no set way to do optimization problems. Really the only things that all the problems have in common is that to solve you need to find a way to relate the unknown quantities that will maximize/minimize another given quantity. It's used just as is says to optimize. If you can relate a unknown quantity(s) in a way which maxs/mins the given quantity.

Say we're maximizing a fence, you've got 200 ft of fence and you're asked to maximize the area. Basically you go through a process find a way to relate the variables (x and y would be used here i guess) to the area. If you maximize the function relating the sides of the fence to the area then you get the dimensions (of the fence in x and y) which would make the area the largest it can ever be with the given constraints.

 

[SpicyRamen]

So for the problem you gave me

I would write-

A= x+y
200= xy

y=200/x

A=x+(200/x)
=1-(200/x^2)
X= (200)^(.5)
= 10*(2)^.5

Y= 10*(2)^.5

Is x and y always the same? And when we're finding the max volume is the base always square(Shape)? Is optimization only used to find the max volume, is there a min volume?

 

[Feldoh]

Area=length x width=x*y
Parameter=2x+2y=200

In the case of a fence you'd maximize the are if you made a square fence so you can fairly easily see that the maximum area is 2500 ft^2 with each side being 50 feet.