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.