(Solved): You are designing a spherical tank, as given in the figure, to hold water for a small village in a ...

You are designing a spherical tank, as given in the figure, to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as \[ V=\pi h^{2} \frac{(\mathrm{SR}-h)}{3} \] where \( V= \) volume \( \left(m^{3}\right) \), \( h= \) depth of water in tank \( (\mathrm{m}) \), and \( R= \) the tank radius \( (\mathrm{m}) \). If \( R=4 \mathrm{~m} \), what depth must the tank be filled to so that it holds \( 50 \mathrm{~m}^{3} \) ? Use three therations of the Newton-Raphson method to determine your answer. Determine the approximate relative error after the third iteration. Note that an initial guess of \( R \) will always converge. The depth of the tank and the approximate relative error after the third iteration are \( \mathrm{m} \) and \%, respectively.