Question $2[65$ marks$]$

The pressure vessel of the Figure Q$2-$a will be used to store some pressurized fluid and should be manufactured using the minimum amount of malerial possible, while holding a total volume $V.$ The amount of material used is proportional to the surface area of the vessel, while the inner radius of the versed should be equal to one third of the outer radius $r_1=\frac{r}{3}$

Figure Q$2-$a$.$ Pressure vessel dir

a$)[15$ marks$]$

Derive expressions for the total surface area and volume of the vessel as functions of $r,O$ and $h.$ In your computations, use the following expressions for the lateral surface area $A_f$ and volume $V_f$ of a frustum $($truncater rnnol

Figure Q$2-$b$.$ Frustum geometry

$A_f=(r+r_1)\sqrt[\phantom{2}]{(r-r_1{)}^2+h_1^2}$

$V_f=\frac{h_1}{3}(r^2+r{r}_1+r_1^2)$

Then, vrite two Python functions named vesselAreae and vesselvolume to compute the total surface area and total volume of the vessel. The functions should both receive the following input arguments:

$\backslash$table$[[$Name$,$Data type,Argument type,Default value,Description$],[$r$,$float,Positional,$-,$Radius of the vessel.$],[$theta$,$float,Keyword,$,$Angle of the truncated cone at the top and bottom of the vessel in radians$],[h,$float,Keyword,$0\mathrm{.}5,$Height of the cylinder$]]$

Function vesselAreae should return the area of the vessel as a float, and vesselvolume should return the volume of the vessel, also as a float Use the tests in the cells following your answer to verify the correctness of your answers

In $[3]$: ~~ YOUR CODE HERE

import math

def vescelarea$(r$ thetra$)$: