Q about making an unconventional bass trap from a new guy

Greetings to all. My name is Jason and I live in Taiwan. I should start by saying this forum is a great resource, and I have already learned a lot from reading the informative posts. Thanks to all those knowledgeable posters willing to share expertise and experiences. It's a wonderful thing you're doing. I am making a bass trap for my home studio. (I will not list my studio dimensions and current acoustic treatment, both for brevity sake and because I don't think it's particularly relevant to this question---but please forgive me if I'm mistaken here) My goal is to make this trap using discarded materials that I have scavenged from the trash for a total project cost that closely approaches $0.00. For the record, this goal has everything to do with environmental concerns, experimental curiosity, and my personal recycling philosophy, and nothing whatsoever to do with budget constraints. Anyway, here's my question: For the bass trap, I plan to use two (2) empty desk drawers, stacked lengthwise, one atop the other, and screwed together. Free, ready-made wood framing like this is quite handy. Each drawer measures 34.5" long X 17" wide X 3.5" deep. The drawer sides are made of 1/2" wood The drawer bottoms are made of 1/8" plywood The drawers will be stood upright and filled to the full 3.5" depth with 4 layers of a material closely resembling cocoa mat. It is fibrous, very rough, and tightly woven. Although (I presume) untested as an absorber, I suspect it should have absorbing properties similar to fiberglass. Based on the cost of my materials (0.00) I have nothing to lose anyway. What I want to know is whether or not keeping the 1/8" plywood (presently the bottom of each drawer) as a backing for the trap would affect it's absobtion/effectiveness. To be clear, once they're stacked, the front of the trap will be 'open', with the absorbant material covered only in fabric, while the back would be 'closed' by the existing 1/8" plywood that is presently the bottom of the drawer. Make sense? Based on what I've read on the subject both here and elsewhere, I suspect the plywood back would make the trap less effective. On the other hand having a solid back makes the job of holding the absorbant material in place that much easier. So I thought I'd ask the experts. Can anyone advise me? Thanks in advance.

Mister Green, I think you covered all points yourself already, including your conclusion about the bottom. Without that bottom you can gain from the cavity behind the cocos mat. In fact your question can be summarized as: Can I use the frame of the drawer as frame of the trap? Well no reason why you couldn't, and what that cocos mat absorbs you should test.