Chapter 6, Problem 9PP

A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The “clock” is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x = f(y) be continuous on the interval [0, b] and assume that the container is formed by rotating the graph of f about the y-axis. Let V denote the volume of water and h the height of the water level at time t.

(a) Determine V as a function of h.

(b) Show that

$\frac{dV}{dt}=\pi {\left[f(h)\right]}^2\frac{dh}{dt}$

(c) Suppose that A is the area of the hole in the bottom of the container. It follows from Torricelli’s Law that the rate of change of the volume of the water is given by

$\frac{dV}{dt}=kA\sqrt{h}$

where k is a negative constant. Determine a formula for the function f such that dh/dt is a constant C. What is the advantage in having dh/dt = C?

FIGURE FOR PROBLEM 11