A certain stretch of a river is monitored for pollutant concentration over the…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
A certain stretch of a river is monitored for pollutant concentration over the 24-hour period that begins at midnight. The concentration in milligrams per liter, denoted C(t), is known to be a continuous function for 0≤t≤24, where t is measured in hours after midnight. Sensor readings report
• C(0)=3.8 mg/L,
• C(12)=5.2 mg/L, and
• C(14)=3.6 mg/L.
Assume no contaminant other than the one being measured is present.
(a) Explain why the equation C(t)=4.5 must have at least one solution for 0<t<12. Provide clear justification based on the given information.
(b) Environmental regulations require that the concentration stay below 4.0 mg/L at all times. Using the data above, determine whether the regulation is certainly met on the interval 0≤t≤14. Justify your conclusion.
(c) Agency officials claim that the concentration exceeded 5.0 mg/L at some time between 8:00 a.m. (t=8) and 10:00 a.m. (t=10). Evaluate whether this claim must necessarily be true and support your answer with reasoning.
(d) In the context of water quality, interpret what your answers in parts (a)–(c) reveal about how the pollutant concentration behaved during the day.