The SodaStream Fizz Soda Maker

My family drinks a LOT of seltzer water. We've been intrigued by a new(ish) product that allows one to carbonate tap water at home. The touted benefits are that it's easy to use, is better for the environment than buying bottled seltzer water, and saves money.

I found the product, and its competition at a couple of local merchants.

Unfortunately, before I was able to take a photo of the replacement cartridge and its price, I was asked to leave by the store management, I guess because of my picture taking,  they thought I was either a corporate spy from Office Depot, or else I was casing the joint for an Oceans 11 style heist later that night...

I was not deterred in my quest, however... , I went online and found the price for the refill cartridge:

Your task is to analyze the manufacturers claim that using their product will be a money saver.  . Please create a clear and precise argument that either supports or refutes the manufacturer's claim.

Your analysis should start with fundamentals, that is the prices of the SodaStream and any additional components and supplies you need. Do not simply rely on someone else's analysis that merely states that the system will pay for itself in x-months.

Of course, if you are taking calculus I expect you will find a way to use a concept or two from calculus in your analysis!