You are working for DOWN THE TOILET COMPANY that makes floats for ABC

commodes. The floating ball has a specific gravity of $0\mathrm{.}6$ and has a radius of $5\mathrm{.}5$ cm$.$ You

are asked to find the depth to which the ball is submerged when floating in water.

The equation that gives the depthx $($unit ofx is m$)$ to which the ball is submerged under

water is given by

$30\mathrm{.}1652+3\mathrm{.}993\backslash$times $104=0$

Solving it exactly would require some effort. However, using numerical techniques such as the

bisection method, we can solve this equation and any other equation of the form$0)(=$xf $.$ Solve

the above equation using the bisection method and do the following. Use initial guesses as$0=$x

and$11\mathrm{.}0=$x $.$

a$)$ Conduct three iterations.

b$)$ Calculate the absolute relative approximate error at the end of each iteration.

Ans$->(--,33\%,20\%)$

c$)$ Find the number of significant digits at least correct at the end of each iteration.

d$)$ How can you use the physics of the problem to develop initial guesses for the bisection

method?

e$)$ Find the solution until two significant digits. Use MATLAB$/$PYTHON$/$Fancy Calculator if

necessary; it may take up to $9$ iterations