Optimization Problem

[DaphneDiamond]

I've been going in circles trying to solve this problem. I'm not very good at minimization and maximization at all. This is for my Calculus AB class by the way.

Question: A 51 meter length of wire is cut into two parts. The first part is shaped into a rectangle that is twice as long as it is wide. The second part is shaped into a square. How much of the original wire is used for each shape if the shape’s combined area is:
a.) Minimized?
b.) Maximized?

Work:
Width of rectangle: x
Length of rectangle: 2x
Area: 2x^2
Perimeter: 2x + 2(2x) = 6x

Square
Length of side: y
Area: y^2
Perimeter: 4y

6x + 4y = 51

Minimized:
Maximized:

That's all my work. I don't know if I've got the right equations or anything, but I have tried to set it up.

 

[LCKurtz]

That's good so far. Now you just need to write down the objective function that you are trying to maximize or minimize. That's the total area, which you know is [MATH]A = 2x^2 + y^2[/MATH], with the constraint that [MATH]6x+4y=51[/MATH]. You can either substitute to make a one variable problem or use LaGrange multipliers if you have studied that. Just keep going...

 

[DaphneDiamond]

Thank you! I completely forgot about LaGrange multipliers. No wonder I didn't know how to solve it.