Together, they make eight questions of the type 'take a pentomino and some rectangle-shaped tiles, and try to fit them in a (bigger) rectangular box'. There are twelve pentominos; so four of them are left, and it's easy to see that it doesn't really make sense to ask the question for I, L and P because they trivially tile a 1x5 or 2x5 rectangle:
It turns out that the Y pentomino (see above) also tiles a rectangle without help of other rectangles, but the solution is a bit trickier and a nice puzzle, certainly doable by hand (that's where the no-computers tag is for). So the actual puzzle here is:
What is the smallest rectangle which can be tiled by Y pentominos?
Rotations and reflections of the pentomino are allowed, so you actually have 8 forms to work with.
One observation/hint (which, as expected, wasn't necessary, but might help future readers who want to solve this puzzle by themselves):
The total area of the rectangle must be divisible by 5. Therefore, at least one of the sides must be divisible by 5 as well.