Problem $5($optional$)$: You and your friends recently finished building the most magnificent treehouse in the neighborhood. The only thing that the treehouse is missing is your favorite drink, Hawaiian Punch. The only way to get up to the treehouse is to climb a ladder, making it difficult to carry up drinks. To fix this problem, you and your friends create a system that pumps Hawaiian Punch through $1$ cm diameter piping from a $5$ L tank on the ground to a Hawaiian Punch self$-$serve station in the treehouse $($the $$station$$ is just the end of the piping that drops Hawaiian Punch into your cup$).$ After taking some measurements, you determine that the height difference from the tank to the Hawaiian Punch pipe outlet is $15$ ft$.$ a$)$ You and your friends don$$t want the Hawaiian Punch to flow too quickly, so you aim for it to exit at a velocity of $0\mathrm{.}2$ m$/$s$.$ What is the power needed to pump the Hawaiian Punch from the tank into your cup assuming there is no friction in the pipe? State any other assumptions. b$)$ What size pump $($in hp$)$ must you use if said pump has an efficiency of $0\mathrm{.}67?$ c$)$ The next day you and your friends decide to hang out in the tree house $(24$ h after the Hawaiian Punch was put in the tank$).$ However, when one of your friends goes to dispense some Hawaiian Punch, you discover that the Hawaiian Punch in the tank has become populated with bacteria. After Googling, you determine that the bacteria populate at a rate proportional to their population, doubling in number every $20$ minutes. How many bacteria are in the tank if the initial number was $1000$ bacteria? State all assumptions. d$)$ To prevent future bacteria growth, you decide to use a refrigeration system in your tank. Hawaiian Punch at room temperature $(25$C$)$ is added to completely fill the tank. How much heat needs to be removed to decrease the temperature of the punch to $3$C$?$ State all assumptions.