The number of gallons of water in a large tank at time (t) minutes is given by the function W (t) = 160,000-8,000t+t². Find the average rate of change of the number of gallons of water in the tank from t=10 to t=10.5 minutes. Interpret the sign of your answer in the context of this problem.

The number of gallons of water in the tank at t=10 is

... W (10) = 160,000 - 10 (8000 - 10) = 80100

The number of gallons of water in the tank at t=10.5 is

... W (10.5) = 160,000 - 10.5 (8000 - 10.5) = 76110.25

The rate of change over the interval is

... (W (10.5) - W (10)) / (10.5 - 10) = (76110.25 - 80100) / (0.5) = - 7979.5

The average rate of change in the number of gallons of water in the tank over the interval is - 7979.5 gal/min.

The sign is negative, so the amount of water is decreasing.