Chapter 6, Problem 19P

You are designing a spherical tank (Fig. P6.19) to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as

$V=\pi h^2\frac{\left[3R-h\right]}{3}$

where $V=$ volume $\left({\text{m}}^3\right),h=$ 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 ${\text{m}}^3$? Use three iterations of the Newton-Raphson method to determine your answer. Determine the approximate relative error after each iteration. Note that an initial guess of R will always converge.

FIGURE P6.19