Example 3: Pollution Rates

Click here to open an associated Mathcad worksheet:

Problem. The graph below gives the approximate amount of solid matter, in tons, dumped daily into Boston Harbor, over the course of a six-year period between  1989  and  1995 ¹. At what rate did the dumping decrease during this period? Why might this rate be misleading?

Solution. Determining values from a bar graph such as this one is an imprecise business – the number of tons being dumped into the harbor each day may change over the course of a year. The graph apparently displays only one-year averages of the daily values. Thus if  T  represents the number of tons per day, we can only give approximate values like  T(during 1989)    130 tons/day  and  T(during 1994)    50 tons/day.

Our calculations of the change in value,  DT = 50 – 130 = –80 tons/day, and the rate of change in value,  DT/Dt = (50 – 130)/(1994 – 1989) = –80/5, or  –16  tons/day per year, are likewise imprecise.

But the rate is misleading for another reason. It is itself an average rate of change over a six year period. The rate of decrease was actually zero during some years (1990–91, 1991–92, 1992–93) and steeper than  –16  tons/day per year in others (1989–90, 1993–94, 1994–95).