Chapter 9.1, Problem 5QC

In Example 7, if the height function were given by h(t) = (4.2 − 0.14t)², at what time would the tank be empty? What does your answer say about the domain of this solution?

Example 7 Flow From a Tank

Imagine a large cylindrical tank with cross-sectional area A. The bottom of the tank has a circular drain with cross-sectional area a. Assume the tank is initially filed with water to a height (in meters) of h(0) = H (Figure 9.7). According to Torricelli’s law, the height of the water t seconds after the drain is opened is described by the differential equation

$h\prime \left(t\right)=-k\sqrt{h}$, where t ≥ 0, $k=\frac{a}{A}\sqrt{2g}$,

Figure 9.7

and g = 9.8 m/s² is the acceleration due to gravity.

1. a. According to the differential equation, is h an increasing or decreasing function of t, for t ≥ 0?
2. b. Verify by substitution that the solution of the initial value problem is $h\left(t\right)={\left(\sqrt{H}-\frac{kt}{2}\right)}^2$.
3. c. Graph the solution for H = 1.44 m, A = 1 m², and a = 0.05 m²,
4. d. After how many seconds is the tank in part (c) empty?