Hello, Olivia!

You made a vailiant effort, but your set-up is off . . .

A 30-cm piece of wire is cut in two.

One piece is bent into the shape of a square.

The other piece is bent into the shape of a rectangle with a length-to-width ratio of 2:1.

The sum of the areas of the square and rectangle is a minimum.

What are the lengths of the two pieces? Let = length of wire for the rectangle.

Then = length of wire for the square.

The rectangle looks like this: Code:

2k
* - - - - - - - *
| |
k | | k
| |
* - - - - - - - *
2k

The perimeter of the rectangle is: .

Its wire is cm long: . . **[1]**

The area of the rectangle is: . . **[2]**

Substitute [1] into [2]: .

The square looks like this: Code:

* - - - - - - *
| |
| |
| | ¼(30-x)
| |
| |
* - - - - - - *
¼(30-x)

The perimeter of the square is: cm,

. . so its side is:

The area of the square is: .

Hence, the total area is: .

And *that* is the function we must minimize . . .