You are designing a spherical tank $($Fig$.$ P$6\mathrm{.}22$ below$)$ to hold water for a small village in a developing country. The volume of liquid it can hold can be computed

as

$V=h^2\frac{(3R-h)}{3}$

where $V=$ volume $(m^3),h=$ depth of water in tank $(m),$ and $R=$ the tank radius $(m).$

FIGURE P$6.22$

If $R=3m,$ what depth must the tank be filled to so that it holds $V=28m^3?$ Use four iterations of the MODIFIED SECANT METHOD. Determine the value of

$h$ for the next iteration, and the percent error between the current and last iteration. Use an initial guesses of $h=R,$ and a perturbation factor $(d)$ of $1\mathrm{.}0E-4.$

For information on the Modified Secant method, see Example $6\mathrm{.}5$ in Chapra's Applied Numerical Methods with Matlab, or look at example calculations in the

excel sheet provided on the Homework Assignment page on Canvas.

Answer:

$h_1=$

$v_1=$

$h_2=$

Error ${?}_{1-2}=$

$v_2=$

$h_3=$

Error ${?}_{2-3}=$

$r_3=$

$h_4$

$($