Problem $11-12($Algo$)$

The bathtub theory of operations management is being promoted as the next breakthrough for global competitiveness. The factory is a bathtub with $50$ gallons of capacity. The drain is the outlet to the market and can output $2\mathrm{.}0$ gallons per hour when wide open. The faucet is the raw material input and can let material in at a rate of four 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.$)$

Maximum rate

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.$)$

Number of hours

b$.$ Suppose that, instead of a faucet, a five$-$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.$)$

Average output rate

gals$/$hr