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 material 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 vessel should be equal to one third of the outer radius $r_1=\frac{r}{3}$

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

d$)[15$ marks$]$ single numerical output argument with the value of the first and second derivative of the surface area respectively. and not using numerical differentiation, since this would lead to inaccurate results. especually for the second derivative.

In $[]$:$=$ YOUR CODE HERE

raise NotImplementedError$()$

In $]$: #Run this cell to test your functions for question $2-$d$)$

from unit$\_$test$=$ import ves$=$elAreaderivativeTest, ves$=$elAreasecondoerivativetest

vesselAreaDerivativetest $($vesselareaDerivative$)$

vesselareasecondDerivativetest $($vesselareasecondoerivative$)$

In $[]$ :

In $[]$ :

e$)[10$ marks$]$ obtained the optimal radius, evaluate the area corresponding to this radius and store it in a variable named areapt

In $C]$:

In $C$ i: $-$ YoUR coDE heRE

raise NotImplementederror$()$

ariun chts cele co test your answers for question $2-$e$)$

import numpy as np

import math

if Dopt not in giobal$=()?$

assert $($t$)$ Voluetonor $($ropt is not definedi?

if iaregopti not in gxabals $)$:

assert $($areappto$).$ The area should be a positive numbent?

print $($iyour answers: for question Z$-$e pass the preliminony checkst"$)$

In $x$ l:

I really need help on part D and E$,$ i only need the answer for both $($Python$)$

Thanks you so much