It's the year $2050$ and while everyone is focused on colonizing Mars, NASA has assigned you the task of working on a mission design to colonize Venus. It turns out that it is possible to establish a floating colony in the clouds of Venus between the freezing vacuum of space and the scorching hot Venusian $($not to be confused with Venetian$)$ surface.

Your task is to compute the mass requirements for both consumables $($food$,$ water, and oxygen$)$ and propellant for the outbound trip from Earth to Venus for a crew of five $(5)$ astronauts.

The spacecraft $($mothership$)$ used to deliver the astronauts to Venus has a dry mass of $60$ mt $($i$.$e$.,$ not including propellant and consumables$)$ and starts in a circular Low$-$Earth Orbit $($LEO$)$ at $300$ km altitude. It needs to transfer to a Low$-$Venusian Orbit $($LVO$)$ at an altitude of $\backslash (\backslash$mathbf$\{500\}\backslash$mathbf$\{$~ k m$\}\backslash )$ where a Venus "Base Camp" is waiting for them with supplies for the rest of their mission.

You are analyzing various possibilities to transfer from LEO to LVO. Refer to the tables at the end of the problem statement for all necessary planetary and ECLSS data and parameters.

You need to analyze and work through ALL of the following options, and answer one short answer.

You MUST solve this problem by hand. Show your work and intermediate steps for full credit.

Option $1$: Interplanetary Hohmann Transfer:

a$.$ What are the total $\backslash (\backslash$Delta $\backslash$mathbf$\{$v$\}\backslash )$ and total Time of Flight $($TOF$)$ for this transfer option from LEO to LVO? Assume that the time spent in Earth's and Venus' SOIs totals to $10$ days $($you still need to compute the interplanetary TOF to get the total TOF$).$ Do not simply state numerical results from examples given in class or in the textbook.

b$.$ How much mass of consumables $($food$,$ water and oxygen$)$ must the astronauts take with them for the duration of this transfer? Add $\backslash (10\backslash \%\backslash )$ to the total mass of consumables for contingencies $($margins$).$

c$.$ If the mothership has a chemical propulsion system with a specific impulse of $450$ s $,$ what are the propellant mass fraction, propellant mass, and total Initial Mass in LEO $($IMLEO$)$ for this option?