Water flows through a -meter section of pipeline carrying a contaminant whose…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Water flows through a 300-meter section of pipeline carrying a contaminant whose concentration varies along the pipe. The concentration C(x), in milligrams per liter, at a point x meters from the pipe’s entrance satisfies C(0)=0.20 and C(300)=0.60. Engineers know that concentration changes smoothly along the pipe.
(1) Environmental regulations require a warning sign wherever the concentration first reaches 0.45 mg/L. Prove that at least one such point exists inside the pipe.
(2) Explain whether the information given is sufficient to conclude that there is a point where the concentration is exactly 0.50 mg/L and located no more than 200 meters from the entrance.
(3) Sensors are to be installed no more than D meters apart so that the data they collect will guarantee detection of every concentration between 0.20 mg/L and 0.60 mg/L at some sensor. Determine an upper bound for D and justify your answer.