Imagine you have the following equations to describe your question:
A = (1*x) + (y^2)
2 + 2x +4y =16
x>1
Where
x: Rectangle Length
y: Square Length and Width
These should be pretty straight forward equations, but it's good to remember that the length of your rectangle has to be the longer of the sides, which is where that inequality comes from.
One of the easiest ways you can calculate the smallest and largest values for A (total area) is by re-arranging you second equation to solve for y, and then plug that into your first equation. This now gives you a single variable function:
A(x) = x + (((7/2)-(1/2)x)^2)
Next, simply showing what your boundaries display should give you a pretty good estimate of what you'll find.
When y ~ 0, and x is max (~7), you'll have:
A = 7
When x ~ 1, and y is max (~3), you'll have:
A = 9
From either point, making the other shapes progressively smaller will give no increase in size to make 9 not the greatest total area.
finding the smallest area can be achieved by taking our equation, finding the derivative, and finding a zero value
A'(x) = (1/2)(x-(11/2)) =
x = 11/2 [5.5]
y =3/4 [0.75]
A = 97/16 [6.025]
