Problem 3. One trend in gastronomy is to put a cloche (lid) over a serving platter, pipe smoke under the cloche, and allow some of this smoke to basically seep into the food and/or beverage, before removing the cloche to serve. We won't be able to precisely model this process (... for now), but with the right simplifications we can come up with some rough estimates. For this scenario, let's assume we have two systems, the air under the cloche, and the food that is being smoked. We will treat the air and the food as "containers" that hold the smoke, so we're only writing a mass balaice on the smoke. We will concern ourselves only with the idea of "transfer" of mass, so we will approximate that there is no chemical reaction going on with the food and the smoke. And of course smoke is not really a "single component," but let's treat is as one for now. The rate (mass per time) at which smoke enters the food could be predicted to be min=k(msatm) where k is a proportionality constant (you should be able to determine the dimensions), msat is the "saturation" mass of smoke, such that no more smoke could possibly enter the food, and m is the mass of smoke in the food at any given point in time. For this scenario, you are asked to come up with a symbolic formula that describes the mass of smoke in the food as a function of time, assuming that the initial mass of smoke in the food is m=0 and the initial mass of smoke in the air under the cloche is m2=M. As you show your work and reasoning to come up with m(t), the mass of smoke in the food as a function of time, please do the following: - Classify this scenario based on the number of systems, components, inputs, outputs, reactivity, and time-dependence. - Explain why you must choose the food and not the air as your system to derive the necessary mathematical model. (Hint: one option here is to try to start with the air as your system, and show how you get a mathematical model that can not be solved by itself.) - State the dimensions on the parameter k. - Solve for mass m as a function of time t, leaving any parameters in your expression. - If we waited for a "very long time" (i.e., set t to infinity), what is the final mass of smoke in the food