GUIDED INSTRUCTION (a) Express the assumption that "the rate of change of the volume of water, V, is proportional to the surface area A. " Use a proportionality constant k (which may be named as "porosity".) Is it a positive or a negative constant? Since this is a crucial step, I will just write the answer for you. dV dt = kA (b) Use the given formulas for the volume and area (displayed above the diagram) to rewrite the equation in (a) as an equation for h dh and dt (We will call this a related rate equation.) (c) Solve the related rate equation from (b) for dh dt and simplify it. Use this equation to explain why the drain rate h' (t) cannot be a constant function unless the filter is completely clogged (that is, unless k = 0.) [Hint: You may use h" (t) to be able to answer this. There [h (t)] and about are other ways, too.] Suppose the water is completely drained at a time t = to. Talk about the limits lim₁→to lim+ →→to [h' (t)]. To answer these, make an assumption that h (t) is a monotonically decreasing continuous function (i.e. water doesn't climb back up!), and that h (to) = 0. Verify your findings analytically by using the related rate equation. (d) Study the concavity of the graph of the function h. We do this so we know whether the water level goes down faster and faster or slower and slower. Is the graph of h always concave up or always concave down or does it have an inflection point? Provide a physical explanation to your finding about the concavity. (e) Sketch a reasonable graph of h (t) and of h' (t) based on your answers in (c) and (d). Certain properties of graph (such as slope at t_0, concavity etc) must be shown clearly and explained. (f) Show that the function h(t) = 2R 1 V k R √√√1 - 1/2 (t - to))