In $2016-2017,$ a system of planets $($called TRAPPIST$-1)$ was discovered $40$ light years away from Earth, as measured by the Earth based observatory.

You are on a team of scientists on Earth who are preparing a mission to explore the system. The spacecraft and payload $(($:$m_3=2\mathrm{.}1{10}^5(kg)\}$ will leave orbit and accelerate to $0\mathrm{.}45$ C and then travel the rest of the distance with a constant velocity.

The spacecraft uses a nuclear reactor to change mass from the fuel into energy for the spacecraft. You have carefully measured the distance to the planet and know the optimal cruising speed.

You need to determine:

$($a$)$ how many days of food to supply the ship with.

$($b$)$ what mass of fuel is needed to reach the optimal cruising speed.

$($You may assume that the acceleration part of the journey does not take any significant time or distance$).$