Environmental scientists measure the concentration (in milligrams per liter) of a…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Environmental scientists measure the concentration (in milligrams per liter) of a chemical in a lake. For 0≤t≤5 days after a spill has been contained, the concentration is modeled by
C(t)=4e−0.3t+1.
To ensure safe conditions for wildlife, officials want the average concentration over the interval [a,5] to be exactly 2mg/L. (Here a is the time, in days after the spill is contained, when monitoring begins.)
Parts:
(a) Explain why C(t) is continuous on [a,5] for any 0≤a≤5, and state the theorem that guarantees the average value is well-defined.
(b) Set up, but do not evaluate, an integral that represents the average concentration of the chemical over [a,5].
(c) Evaluate the integral from part (b) and obtain an expression for the average concentration in terms of a.
(d) Solve for the value of a that makes the average concentration equal to 2mg/L. Show each algebraic step and verify that your solution lies in the interval [0,5].
(e) Interpret the practical meaning of your result from part (d) for the environmental monitoring plan.