Problems

$(3$ points$)$ Estra, who you've come to learn is the alien ship's navigator, is planning their route and is concerned about their fuel levels for their journey home. The alien ship uses a mysterious fuel called orsh, a substance unfamiliar to humanity.

Estra estimates that the regular functions of their ship while it sits at UTM $($lights$,$ communications, hydroponic fruit gardens, and so on$)$ consume orsh at a rate between $4000$ and $5000$ litres per day. The ship's fuel gauge says there are currently $300000$ litres of orsh on board, but Estra tells you that the fuel gauge's reading can be off by as much as $3000$ litres $($they$\text{'}$ve been meaning to get it fixed, but it's so hard to find a good mechanic...$).$

$($a$)(2$ points$)$ Assuming Estra's estimates are correct, and accounting for the potential error from the orsh gauge, what is the range of possible amounts of orsh that might be left in the ship's tank after two weeks of normal operation?

State your final answer as an interval $($like $a,b).$ Explain your answer, and justify the use of any theorems you use.

$($b$)(1$ point$)$ Estra believes the journey to their home planet will require no less than $200000$ litres of orsh. Again, assuming their estimate for the ship's orsh consumption is correct $($and still accounting for the gauge's error$),$ how many more full days can the ship remain at UTM before it no longer has enough orsh to make the journey home?

State your final answer as an integer. Explain your answer, and justify the use of any theorems you use.

$(3$ points$)$ Estra, who you've come to learn is the alien ship's navigator, is planning their route and is concerned about their fuel levels for their journey home. The alien ship uses a mysterious fuel called orsh, a substance unfamiliar to humanity.

Estra estimates that the regular functions of their ship while it sits at UTM $($lights$,$ communications, hydroponic fruit gardens, and so on$)$ consume orsh at a rate between $4000$ and $5000$ litres per day. The ship's fuel gauge says there are currently $300000$ litres of orsh on board, but Estra tells you that the fuel gauge's reading can be off by as much as $3000$ litres $($they$\text{'}$ve been meaning to get it fixed, but it's so hard to find a good mechanic...$).$

$($a$)(2$ points$)$ Assuming Estra's estimates are correct, and accounting for the potential error from the orsh gauge, what is the range of possible amounts of orsh that might be left in the ship's tank after two weeks of normal operation?

State your final answer as an interval $($like $a,b).$ Explain your answer, and justify the use of any theorems you use.

$($b$)(1$ point$)$ Estra believes the journey to their home planet will require no less than $200000$ litres of orsh. Again, assuming their estimate for the ship's orsh consumption is correct $($and still accounting for the gauge's error$),$ how many more full days can the ship remain at UTM before it no longer has enough orsh to make the journey home?

State your final answer as an integer. Explain your answer, and justify the use of any theorems you use.

$2.(3$ points$)$ The aliens want to stay longer than your estimate from Q$1($b$)$ allows, so their chief scientist Genly has come up with a way of converting water into orsh. He says that by applying a great deal of pressure to water, the aliens can obtain many litres of orsh from a single litre of water, with the exact amount varying as a function of the pressure.

Genly's first version of this process produces orsh according to the following function, where $G(P)$ is the number of litres of orsh produced by applying a pressure $P>0($measured in kilopascals, kP$)$ to a litre of water:

$G(P)=\frac{5P^2}{10+(P-10{)}^2}$

The aliens can apply very large amounts of pressure to the water, but need your help to determine what to do$.$

$($a$)($Warm$-$up$!$ Do not submit this part to Crowdmark.$)$ Check that $G(10)=50.$ What are the units for this quantity? Compute the value of $G$ for some other values of $P.$ What do you think is happening for large values of $P?$

$($b$)(1$ point$)$ Show with a limit that increasing the pressure $P$ to very, very large levels doesn't produce more orsh per litre of water.

$($c$)(2$ points$)$ Find the ideal pressure $P_i$ at which this process is most efficient $($i$.$e$,.$ at which it produces the largest quantity of orsh per litre of water$).$ How much orsh is produced $($per litre of water$)$ at the pressure $P_i?$

Clearly state your two answers at the beginning of your solution, and show your work below. Don't forget to justify why your answer is where this maximum occurs.

$(2$ points$)$ Genly's first version of the process was good, but he is confident he can do better.

He spends another day refining his ideas,