Geometry
# Quadrilaterals

Suppose $ABCD$ is an isosceles trapezoid with bases $AB$ and $CD$ and sides $AD$ and $BC$ such that $|CD| \gt |AB|.$ Also suppose that $|CD| = |AC|$ and that the altitude of the trapezoid is equal to $|AB|.$

If $\dfrac{|AB|}{|CD|} = \dfrac{a}{b},$ where $a$ and $b$ are positive coprime integers, then find $\large a^{b}.$

The numbers $3,4,$ and $6$ denote the areas enclosed by their respective triangles.

What is the area of the yellow region?

I want to make a litter box for my newly adopted pet kitten Admiral. The box is a cuboid with the top removed. I want the volume of the box to be 32 but the surface area to be minimized.

What is the minimum internal surface area I can achieve?