Please solve the following problem with Python: Problem $3.$C$5$ described by Fig. P$3.$C$5.$

The exact expression is given in Prob. $3\mathrm{.}64$ for the angle of twist of the solid tapered shaft $AB$ when a torque T is applied as shown. Derive an approximate expression for the angle of twist by replacing the tapered shaft by $n$ cylindrical shafts of equal length and of radius $r_i=(n+i-\frac{1}{2})(\frac{c}{n}),$ where $i=1,2,$dots,$n.$ Using for $T,L,G,$ and $c$ values of your choice, determine the percentage error in the approximate expression when $($a$)n=4,($b$)n=8,($c$)n=20,($d$)n=100.$

$3.$C$5$ The exact expression is given in Prob. $3\mathrm{.}60$ for the angle of twist of the solid tapered shaft $AB$

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when a torque $T$ is applied as shown. Derive an approximate expression for the angle of twist by replacing the tapered shaft by $n$ cylindrical shafts of equal length and of radius $r_i=(n+i-\frac{1}{2})(\frac{c}{n}),$ where $i=1,2,$dots,$n.$ Using for $T,L,G,$ and $c$ values of your choice, determine the percentage error in the approximate expression when $($a$)n=4,($b$)n=8,($c$)n=20,($d$)n=100.$

Fig. P$3.$C$5$

$3\mathrm{.}60$ A torque $T$ is applied as shown to a solid tapered shaft $AB.$ Show by integration that the angle of twist at $A$ is

$($a$)-3\mathrm{.}282\%.($b$)-0\mathrm{.}853\%.$

$($c$)-0\mathrm{.}138\%.($d$)-0\mathrm{.}00554\%.$

$=\frac{7TL}{12G{c}^4}$