Q$2$: A hollow spherical shell with an inner radius $r_i=0\mathrm{.}1m$ and outer radius $r_0=0\mathrm{.}2m$ is experiencing steady$-$state, one$-$dimensional radial conduction with uniform volumetric internal heat generation $(q^{\text{'}\text{'}\text{'}}).$ The material has constant thermal conductivity $k=15\frac{W}{m}.K.$ The outer surface at $r_i$ is cooled by a fluid at $T_{}=25C,$ with a convection heat transfer coefficient $h=300\frac{W}{m^2}.$ K$.$ The outer surface temperature is measured to be $T(r_0)=75C.$ The inner surface at $r_0$ is insulated $($no heat flux at $r_0).$ The temperature distribution within the shell has the form:

$T(r)=a+\frac{b}{r}+c{r}^2$

Required:

i$)$ Determine the heat generation rate $q^{\text{'}\text{'}\text{'}}$

ii$)$ Solve for the coefficients $a,b,$ and c in the temperature distribution equation.

iii$)$ Consider conditions for which the convection coefficient is halved, but $q^{\text{'}\text{'}\text{'}}$ remains unchanged.

Determine the new values of $a,b$ and $c.$

iv$-$ Under conditions for which $q^{\text{'}\text{'}\text{'}}$ is doubled, and the convection coefficient remains unchanged $(h=$

$($:$300\frac{W}{m^2}.K$

$*$numerical solution with steps please