A town is using water from a reservoir that is being refille

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? ___ gallons per hour
b.) Name the earliest time at which the overall rate of usage is 150 gallons per hour.  t= ____ hours after midnight
c.) Water flows into the reservoir at a constant rate of 100 gallons per hour. Suppose first that the reservoir is empty and 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?

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 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? ____ gallons

Solution

The graph of inflow is a line through the origin with slope 150. In order that there never be shortage , the inflow graph would have to be above the usage graph at all time. There is an interval on which the usage graph is above the inflow graph.

If the rate of inflow remains 100 gallons per hour, the graph of inflow is always a line with slope 100.