Problem $1($L$12/13)[40\backslash \%]$

You are working on the preliminary design of a new Earth$-$orbiting small satellite which will be in a $500$ km altitude near$-$circular orbit when it completes its mission. The mass of the satellite is $\backslash ($ m$=100\backslash$mathrm$\{$~kg$\}\backslash ),$ and its drag coefficient and drag reference areas are $\backslash ($ C$\_\{$D$\}=1\mathrm{.}0\backslash )$ and $\backslash ($ A$=0\mathrm{.}75\backslash$mathrm$\{$~m$\}^\{2\}\backslash )$ respectively. The table below shows the average atmospheric density across $50$ km altitude increments $($i$.$e$.,$ assume the atmospheric density is constant at the given average value in each altitude range$).$

Assume that when the spacecraft reaches an altitude of $300$ km it will rapidly deorbit $($i$.$e$.,$ near$-$instantaneously relative to the time it took to go from $500$ km to $300$ km $).$ Does the spacecraft comply with the FCC$\text{'}$s $5-$year rule as designed? If so:

$1.$ Determine the maximum amount of mass that can be added to the spacecraft and still have it comply with the rule.

If not:

$1.$ Determine the minimum amount of mass that will have to be removed from the spacecraft in order to make it comply with the rule.

What are the implications of these mass additions$/$subtractions on the overall spacecraft design? Assume that the gravitational parameter of the Earth is $\backslash (\backslash$mu$=3\mathrm{.}986\backslash$times $\backslash )\backslash (10^\{14\}\backslash$mathrm$\{$~m$\}^\{3\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ and its radius is $\backslash (6,378\backslash$mathrm$\{$~km$\}\backslash ).$