A manufacturing process releases a pollutant into the air at a non-negative
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A manufacturing process releases a pollutant into the air at a non-negative, decreasing rate p(t) (in grams per hour) for t≥0. Let A(t)=∫0tp(s)ds denote the total mass of pollutant emitted during the first t hours. Assume p is continuous and p(0)>0. For a fixed target mass M>0, let T(M) be the unique time at which A(T(M))=M.
(a) Explain why T(M) is well-defined and increasing as a function of M.
(b) Without performing any calculations, compare T(2M) to 2T(M). Determine whether T(2M) is less than, equal to, or greater than 2T(M), and provide a clear verbal justification based on the given properties of p.