Chapter $5$:

$5\mathrm{.}2$

$5\mathrm{.}2$ Determine the real root of $f(x)=5x^3-5x^2+6x-2$ :

$($a$)$ Graphically.

$($b$)$ Using bisection to locate the root. Employ initial guesses of $x_i=0$ and $x_{if}=1$ and iterate until the estimated error ${}_a$ falls below a level of ${}_s=10\%.$

For a$)$ use Matlab. For b$)$ show your calculation for at least one step per calculation type $($error$,$ update, function evaluation, etc$)$

$5\mathrm{.}17$

$5\mathrm{.}17$ You are designing a spherical tank $($Fig$.$ P$5\mathrm{.}17)$ 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$5.17$

If $R=3m,$ to what depth must the tank be filled so that it holds $30m^3?$ Use three iterations of the false$-$position method to determine your answer. Determine the approximate relative error after each iteration. Employ initial guesses of $0$ and $R.$