Determine the smallest area these figures can get

Question

Of 400 cm steel wire, one should construct a square and a rectangle where the longer side of the rectangle is 20 cm longer than the shorter side of the same rectangle. Determine the smallest possible area that the two figures can get.

Answer 1

I see 3 unknowns:

x = side of the square

y and z = sides of the rectangle

And we are told that z = y + 20

We also know that 4x + 2y + 2z = 400

Use the 1st equation to sub in the 2nd:

4x + 2y + 2(y + 20) = 400

4x + 4y = 360

x = 90 - y

The total area is given by A = x² + y(y + 20) = x² + y² + 20y

Replace x with 90 - y:

A = (90 - y)² + y² + 20y

Expand and simplify:

A = 8100 - 180y + y² + y² + 20y = 2y² - 160y + 8100

To get a minimum, take the 1st derivative and set = to 0:

A' = 4y - 160 = 0

4y = 160

y = 40

SO.......

the rectangle is 40 X 60, and the square is 10√2 X 10√2 (total length 80 + 120 + 200 = 400)