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I am doing a school project and was wondering if someone could tell me the cost to manufacture the directtv cable box.
It's hard to say for sure, since much of that I'm sure is proprietary / trade secret information for the manufacturers of the boxes. I'm having a hard time myself trying to find this information. For the most part, since the components are typically being sourced from Taiwan, Korea or China and mass produced, you're usually paying for a unit that likely cost no more than $160 a receiver to make. System-on-chip designs are very cheap to make in bulk, don't require a lot of power, and assuming the figure I mentioned earlier is for a DVR, usually no more than $50-60 for a hard drive. Realistically, smartphones are doing far more in a smaller amount of space, which is why you find phones costing $700 but are known to only cost $250-300 to actually fully produce for a rock-solid, well-built, phone.
Much of the cost of the boxes you actually pay for, comes from licensing of DRM technologies to allow the box to decode encrypted TV signals, and to also output an encrypted TV signal if using HDMI (there's this thing called HDCP). You've also got to hire programmers who can design software for the boxes, go through Quality Control both at assembly and at programming, and besides a bunch of other steps you likely have a little bit of padding on top so the company making the boxes can bring in some profit. There are a ton of things that add to the overall cost, but otherwise, overall you'll find manufacturing is cheap compared to dealing with patents and licensing.
Granted, don't quote me on this one for your project since I'm speculating, but hopefully this helps guide your research for your project.