A town is using water from a reservoir that is being refilled with a system of aqueducts. The graph below shows the total water drawn from the reservoir over the course of a typical day, starting at midnight.

a) What is the overall rate of usage at 5 am?

b) Name the earliest time at which the overall rate of usage is 15,150 gallons per hour.

c) Water flows into the reservoir at a constant rate of 100 gallons per hour. Suppose first that the reservoir is empty. Add the graph of Total Flow In (gallons) to the Usage graph above. During how many one-hour intervals is water flowing out at the same rate it is flowing in?

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d) Again, if the reservoir is empty at midnight and water flows in at a constant rate of 100 gallons per hour, then there will not be enough water available to serve the town during this 24-hour period. When does the water shortage begin?

at \( t = \) hours after midnight

e) Again, if water flows in at a constant rate of 100 gallons per hour, what is the smallest amount of water needed in the reservoir at midnight so that the town gets all the water it needs during this 24-hour period? (HINT: How could you change the Inflow graph so that there is never a shortage?)

\( X_e \) gallons

Water usage rate: 400 gallons/hr at 5 am, reaches 1500 gallons/hr at 10 am. Water shortages start 14 hours after midnight. Minimum of 1400 gallons are needed in the reservoir at midnight for a 24-hour water supply.

a.) The overall rate of usage at 5 am is 400 gallons per hour.

b.) The earliest time at which the overall rate of usage is 1500 gallons per hour is 10 am.

c.) If the reservoir is empty and water flows into the reservoir at a constant rate of 100 gallons per hour, then there are 6 one-hour intervals during which water is flowing out at the same rate it is flowing in. These intervals are 1-2, 2-3, 3-4, 8-9, 9-10, and 10-11.

d.) If the reservoir is empty at midnight and water flows in at a constant rate of 100 gallons per hour, then the water shortage begins at t=14 hours after midnight.

e.) The smallest amount of water needed in the reservoir at midnight so that the town gets all the water it needs during this 24-hour period is 1400 gallons.

To see this, we can add the graph of Total Flow In (gallons) to the Usage graph above. The graph of Total Flow In is a horizontal line at 100 gallons per hour.

The two graphs intersect at t=14 hours, which means that at this time the amount of water in the reservoir is 1400 gallons. If the amount of water in the reservoir is at least 1400 gallons at midnight, then the town will have enough water to last until the next day.

If we change the Inflow graph so that it is always above the Usage graph, then there will never be a shortage of water. This is because the inflow of water will always be greater than the outflow of water.

To do this, we could make the Inflow graph a straight line that is parallel to the Usage graph and that is 100 gallons per hour above the Usage graph.

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