Question

A cube of surface area X is sliced into two rectangular prisms.
One of the prisms has surface area 1/2X. What is the surface area
of the other prism?

Answer 1

each side of the cube has area x/6, so each edge has length √(x/6)
one prism, with height h has area 2(x/6) + 4h√(x/6)
So, we know that
x/3 + 4h√(x/6) = x/2
h = √(x/6)/4

The other prism, with height √(x/6)-h has area
x/3 + 4(√(x/6)-h)√(x/6)
= x/3 + 4(√(x/6)-√(x/6)/4)√(x/6)
= x/3 + 4(3/4 √(x/6))√(x/6)
= x/3 + 3(√(x/6))^2
= x/3 + 3(x/6)
= x/3 + x/2
= 5x/6

or, a little less brute-force, ...
Slicing the cube in two adds two more square faces
Each of the original faces has area x/6
Now we have a total area of
x + x/3 = 4x/3
one prism has area x/2, so the other has area
4x/3 - x/2 = 5x/6

Answer 2

nice