$19.$ A single cylinder double acting steam engine develops $150$ kW at a mean speed of $100$ r$.$p$.$m$.$ The coefficient of fluctuation of energy is $0\mathrm{.}08$ and the fluctuation of speed is $\backslash (\backslash$pm $3\backslash \%\backslash )$ of mean speed. If the mean diameter of the flywheel rim is $2$ metres and the hub and spokes provide $7$ percent of the rotational inertia of the wheel, find the mass of the flywheel and cross$-$sectional area of the rim. Assume the density of the flywheel material $($which is cast iron$)$ as $\backslash (7400\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$