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The bathtub theory of operations management is being promoted as the next breakthrough for global competitiveness. The factory is a bathtub with $60$ gallons of capacity. The drain is the outlet to the market and can output $5\mathrm{.}5$ gallons per hour when wide open. The faucet is the raw material input and can let material in at a rate of seven gallons per hour. Now, to test your comprehension of the intricacies of operations $($assume the bathtub is empty to begin with$)$:

a$-1.$ What is the maximum rate $($gallons per hour$)$ at which the market can be served if all valves are set to maximum? $($Round your answer to $1$ decimal place.$)$

Answer is complete and correct.

Maximum rate

$5\mathrm{.}5$

gals$/$hr

a$-2.$ If the faucet is running at maximum and the drain is wide open, in how many hours will the bathtub completely fill and start to overflow? $($Round your answer to $1$ decimal place.$)$

Answer is complete but not entirely correct.

Number of hours

b$.$ Suppose that, instead of a faucet, a eight$-$gallon container is used for filling the bathtub $($assume a full container is next to the tub to begin with$)$; it takes two hours to refill the container and return it to the bathtub. What is the average output rate $($gallons per hour$)?($Round your answer to $1$ decimal place.$)$

Answer is complete but not entirely correct.

Average output rate

$5\mathrm{.}5$

gals$/$hr