T (hours) 0 1 5 8 10 G (t) (thousands of gallons) 22 18.5 19.5 22 24
A gasoline storage tank is filled by a pipeline from a refinery. At the same time, gasoline flows from the tank into trucks that will make deliveries to gasoline stations. The amount of gasoline in the storage tank at time t is given by the twice-differentiable function G, where t is measured in hours and 0≤t≤10. Values of G(t), in thousands of gallons, at selected times t are given in the table above. It is known that G′′(t)>0 for 5≤t<10.
(a) Use the data in the table to estimate the rate of change of the amount of gasoline in the storage tank at time t=3 hours. Show the computations that lead to your answer. Indicate units of measure.
(b) For 0
(c) It is known that G′(5)=0.5. Use the locally linear approximation for G at time t=5 to approximate the amount of gasoline in the storage tank at time t=7. Is this approximation an overestimate or an underestimate for the actual amount of gasoline in the storage tank at time t=7 ? Give a reason for your answer.
(d) The rate at which gasoline flows out of the storage tank into trucks at time t can be modeled by the function R defined by R(t)=100t/(t^2 +4), where t is measured in hours, and R(t) is measured in thousands of gallons. Based on the model, at what time t, for 0≤t≤10, is the rate at which gasoline flows out of the storage tank an absolute maximum? Justify your answer.