As shown in the notes, the velocity as a function of time $(t)$ for a object of mass $m$ falling due to gravity $(g)$ and subject to aerodynamic coefficient $c_d)$ is given by

$v(t)=\sqrt[\phantom{2}]{\frac{gm}{c_d}}tanh(\sqrt[\phantom{2}]{\frac{g{c}_d}{m}}t)$

where tanh is the hyperbolic tangent. From an experiment for an object with mass $m=74kg,$ a velocity of $40\frac{m}{s}$ is measured after $5\mathrm{.}5s$ of free fall. Given this data, it is possible to determine the drag coefficient by solving the nonlinear equation above for the given data and with $g=9\mathrm{.}81\frac{m}{s^2}.$

For this problem, you will determine the value of $c_d$ for the given data using the bisection method by hand calculation. You can use your calculator $($if it has the tanh$)$ or a MATLAB anonymous function to calculate the values of $f(c_d).$ First, put the nonlinear function in the form $f(c_d)=0.$ Start with an initial bracket for $c_d$ between $0\mathrm{.}2$ and $0\mathrm{.}3.$ Create a table similar to Table $1$ of Chapter $03\mathrm{.}03$ in the HNM text. Iterate until the approximate relative error falls below $0\mathrm{.}01\%.$ If you use MATLAB, show the form of the anonymous function $($you can write it on paper, you don't need the output from MATLAB$).$ The relative approximation error $lo{n}_a$ is defined by

$lo{n}_a=|\frac{c_d^{\text{n}\text{e}\text{w}}-c_d^{\text{o}\text{l}\text{d}}}{c_d^{\text{n}\text{e}\text{w}}}|100$