Quickly cooling lithium batteries after charging is a key aspect of extending battery life. Recently Tesla announced a new battery form factor, the $4680,$ which has a diameter of $46$ mm and length of $80$ mm $.$ This is significantly bigger than the previous generation battery, the $2170,$ which has a diameter of $21$ mm and length of $70$ mm $.$ As an engineer, you are curious about the cooling time implications for this larger battery.

Assume that after charging, the cylindrical battery cell is initially at $\backslash (80^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The battery is cooled via convection $($circulating fluid within the battery pack$)$ with fluid at $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and h of $\backslash (50\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$mathrm$\{$~K$\}\backslash ).$ The battery cell has a thermal conductivity of $\backslash (1\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$mK$\}\backslash ),$ a density of $\backslash (2500\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ),$ and a heat capacity of $\backslash (1000\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kgK$\}\backslash ).$

a$)$ How long does it take the center temperature of the smaller $2170$ cell to cool to $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )?$

b$)$ How many times longer does it take the center temperature of the larger $4680$ cell to cool to $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )?$

of What are the likely implications for Tesla?