Chapter 3.2, Problem 33E

Time Drips By The clepsydra, or water clock, was a device that the ancient Egyptians, Greeks, Romans, and Chinese used to measure the passage of time by observing the change in the height of water that was permitted to flow out of a small hole in the bottom of a container or tank.

1. (a) Suppose a tank is made of glass and has the shape of a right- circular cylinder of radius 1 ft. Assume that h(0) = 2 ft corresponds to water filled to the top of the tank, a hole in the bottom is circular with radius $\frac{1}{32}\text{in}$., g = 32 ft/s², and c = 0.6. Use the differential equation in Problem 12 to find the height h(t) of the water.
2. (b) For the tank in part (a), how far up from its bottom should a mark be made on its side, as shown in Figure 3.2.9, that corresponds to the passage of one hour? Next determine where to place the marks corresponding to the passage of 2 hr, 3 hr, …, 12hr. Explain why these marks are not evenly spaced.