Water clocks were used in Egypt and other parts of the ancient world to monitor the daily passage of time. In the version shown in the figure below, an open container of varying radius R(z) drains through a small opening of diameter d0 in its bottom. The mean velocity at the opening is vo(t) and the water level is H(t). If the dimensions are just right, dH/dt will be constant. With each hour yielding the same decrease in H, a linear scale can be used to tell the time. You are asked to design such a clock. (a) Use conservation of mass to relate dH/dt to vo, do, and R(H). (b) Assume that vo follows Torricellis law, vo(t) = p2gH(t), and let C = |dH/dt|be the rate of change in the water level. Identify the function R(z) that will make C constant. (c) Express the initial water volume V0 as a function of do, C, and the initial height H0. (d) The clock should not be allowed to drain completely (why?). Let Hf be the water level after operation for a time tf . For a 12-hour clock that would fit in a room, choose reasonable values for H0Hf and thus C.