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.

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