You are designing a spherical tank (Fig. P6.22) to hold water for a small village in a

Chapter 6, Problem 6.22

(choose chapter or problem)

You are designing a spherical tank (Fig. P6.22) to hold water for a small village in a developing country. The volume of liquid it can hold can be compute as

\(V=\pi h^{2} \frac{[3 R-h]}{3}\)

where \(V=\text { volume }\left[\mathrm{m}^{3}\right]\), h = depth of water in tank [m], and R = the tank radius [m].

If R = 3 m, what depth must the tank be filled to so that it holds \(30 \mathrm{~m}^{3}\)? Use three iterations of the most efficient numerical method possible to determine your answer. Determine the approximate relative error after each iteration. Also, provide justification for your choice of method. Extra information:

(a) For bracketing methods, initial guesses of 0 and R will bracket a single root for this example.

(b) For open methods, an initial guess of R will always converge.